UNIT

7 GEOMETRY AND
MEASUREMENTS

Unit outcomes
After Completing this unit, you Should be able to:
 understand basic concepts about right angled triangles.
 apply some important theorems on right angled triangles.
 know basic principles of trigonometric ratios.
 know different types of pyramid and common parts of
them.

Introduction
In this unit you will in detail learn about the basic properties of right angled
triangles, by using two theorems on this triangle. You will also learns about a
new concept that is very important in the field of mathematics known as
trigonometric ratios and their real life application to solve simple problems. In
addition to this you will also learn a solid objects known as pyramids and cone
and their basic parts.

7.1 Theorems on the Right Angled Triangle

In your earlier classes you have learnt many things about triangles. By now, you
do have relatively efficient knowledge on some of the properties of triangles in
general. In this sub topic we will give special attention to the properties of right
angled triangles and the theorems related to them.
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Right angled triangles have special properties as compared to other types of
triangles. Due to their special nature they have interesting properties to deal with.
There are some theorems and their converses that deal with the properties of right
angled triangles.

Group Work 7.1

1. Name the altitudes drawn from the right angle to the hypotenuse of
the given right angled triangles.
a) b) c)
Q A R B
I

R S H G
J
C
Figure 7.1
p

2.
C In Figure 7.2 To the left of the unknown
quantities
∆ ABC ∆ ADC ∆ BDC
a Hypotenuse
b
h leg
leg
α β
A e B
D f
c
Figure 7.2
3. In Figure 7.2 above, find three similar triangles.
4. In Figure 7.2 above ∆ CAB ∼ ∆ DAC. Why?

Historical note
There are no known records of the exact date
or place of Euclid's birth, and little is known
about his personal life. Euclid is often referred
to as the "Father of Geometry." He wrote the
most enduring mathematical work of all time,
the Elements, a13volume work. The
Elements cover plane geometry, arithmetic
and number theory, irrational numbers, and
solid geometry.
Figure 7.3 Euclid
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7.1.1 Euclid's Theorem and Its Converse
In Figure 7.2 above the altitude CD of ∆ABC divides the triangle in to two right
angled triangles: ∆ADC and ∆BDC. You can identify three right angled triangles
(∆ABC, ∆ADC and ∆BDC). If you consider the side correspondence of the three
triangles as indicated in Table 7.1 below, it is possible to show a similarity
between the triangles.
Table 7.1
∆ ABC ∆ ADC ∆ BDC
Hypotenuse c b a
leg a h f
leg b e h

The AA similarity theorem could be used to show that:

1. ∆ DBC ~ ∆ DCA
2. ∆ ABC ~ ∆ CBD
3. ∆ CAB ~ ∆ DAC
From similarity (2) you get the following proportion:
AB BC
=
CB BD
c a
⇒ =
a f
⇒ a 2 = cf
and from similarity (3) you get the following proportion:
CA AB
=
DA AC
b c
⇒ =
e b
b 2 = ec
These relations are known as Euclid's Theorem.
From the above discussion, you can state the Euclid's theorem and its converse.

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Theorem 7.1: (Euclid's Theorem)

In a right angled triangle with an altitude to the hypotenuse, the
square of the length of each leg of the triangle is equal to the
product of the hypotenuse and the length of the adjacent
segment into which the altitude divides the hypotenuse:
Symbolically: 1. (BC)2 = AB × BD C
Or a = c × f
2
b a
2. (AC) =AB × DA
2

Or b2 = c × e A e B
cD f
Figure 7.4

Example1: In Figure 7.5 to the right,

∆ABC is a right angled C

triangle with CD the a b

altitude on the hypotenuse.

B A
Determine the lengths of 12 cD 3
Figure 7.5
AC and BC if
AD= 3cm and DB= 12cm.
Solution:
(AC)2= (AB) × (AD) .............. Euclid's Theorem
(AC)2= (15cm) × (3cm) = 45cm2… AB = BD +AD

AC= 45 cm 2

AC = 3 5cm
(BC)2 = (AB) × (BD) ............ Euclid's Theorem
(BC)2= (15cm) × (12cm)
( BC) = 180cm 2

BC = 6 5cm

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Theorem 7.2: (Converse of Euclid's Theorem)

In a triangle if the square of each shorter side of the triangle is
equal to the product of the length of the longest side of the
triangle and the adjacent segment into which the altitude to
the longest side divides this side, then the triangle is right
angled:
Symbolically: 1. a2 = cf and
2. b2= ce if and only if ∆ABC is right angled.
C

b a
h

A e f B
cD
Figure 7.6

You can combine the theorem of Euclid’s and its converse as

follows:

∆ABC with side lengths a, b, c and h the length of the altitude

to the longest side and e, f the lengths of the segments into
which the altitude divide the longest side and adjacent to the
sides with lengths a and b respectively is right angled if and
only if a2=cf and b2 =ce.
Symbolically, ∆ABC is right angled.
If and only if a2 = cf and if and only if b2 = ce.
C

b a

h
A e f B
cD
Figure 7.7

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Example 2: In Figure 7.8 to the right,
C
AD = 4cm, DB = 12cm,
b a
AC = 8cm and BC = 8 3cm
and m (∠ADC) = 900. A B
e cD f
Is ∆ABC a right angled? Figure 7.8
Solution:

a. (BC)2= (BD) × (BA) ........... Theorem 7.1

(8 3cm) 2 = (12cm) × (BA)
192cm2= (12cm) (BA)
192cm 2
BA =
12cm
BA = 16cm
Thus (AB) × (DB) = (16cm) × (12cm)
= 192cm2
Hence (BC)2 = (BD)× (BA)
b. (AC)2 = (8cm)2 = 64cm2 = b2
(AD) × (AB) = (16cm) × (4cm)
= 64cm2 = ce
Hence b2 = ec
Therefore from (a) and (b) and by theorem 7.2, ∆ABC is a right angled
triangle, where the right angle is at C.

Example 3: In Figure 7.9 below, AC = 3 13cm, BC = 2 13cm,

AB = 13cm and DB=4cm. Is ∆ ABC a right angled?
C

b a

A e f B
cD
Figure 7.9

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Solution:

a. (BC)2 = (BD) ×(AB) ........... Theorem 7.1

Now (BC) 2 = (2 13cm) 2 = 4 × 13 = 52cm 2 =a 2
(AB)×(BD)=(13cm) × (4cm)=52cm2=fc
Therefore, a2=fc
b. (AC)2=(AD) × (AB) . . . . . . . Theorem 7.1
( )
Now (AC)2= 3 13cm 2 = 9 × 13 = 117cm 2 =b 2
(AB) × (AD) = 13cm × 9cm = 117cm 2 = ec
Therefore, b2 = ec
Therefore, from (a) and (b) above and by theorem 7.2 ABC is a right
angled triangle.

Exercise 7A
1. In Figure 7.10 to the right, ∆ ACB is C

a right triangle with the right angle at

C and CD ⊥ AB where D is on AB .
A B
Find the lengths of AC and 6cm D 12cm
BC, if AD = 6cm and DB = 12cm. Figure 7.10

2. In Figure 7.11 below, ∆ABC is a right triangle. m(∠ABC)= 900, BD is

the altitude to the hypotenuse AC of ∆ABC. Find the values of the
variables.
A
C
4cm
D 8 cm
y cm
9cm
z cm D
A B x cm
9 cm
(a) x cm

Figure 7.11 B
y cm C
(b)
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3. In Figure 7.12 to the right, ∆ABC is right angled at B, B
BD ⊥ AC, BE = BC, BE = 6cm, AC = 12cm.
Find a.
b.
c.
A C
E D
12 cm
Figure 7.12
C
4. In Figure 7.13, AD = 3.2 cm,
DB = 1.8 cm AC = 4cm and
BC = 3 cm. Is ∆ABC a right angled?

A B
D
Figure 7.13

Challenge problems
5. In Figure 7.14 below, ABC is a semicircle with center at O. BD ⊥ AC
such that BD = 8cm and BC = 10cm.
B
Find a.
b.
c. •
A D C
d. O
Figure 7.14

6. In Figure 7.15 below, O, is the center of the semicircle ABC.

BD ⊥ AC, DO = 3cm and BD = 6cm. Find the radius of the circle.

A •
D O C

Figure 7.15

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7.1.2 The Pythagoras' Theorem and Its Converse

Historical note
Early writers agree that Pythagoras was born on
Samos the Greek island in the eastern Aegean
Sea. Pythagoras was a Greek religious leader and
a philosopher who made developments in
astronomy, mathematics, and music theories.
Figure 7.16 Pythagoras

Group work 7.2

1. Verify the Pythagorean property by counting the small squares in the
diagrams.

(a) Figure 7.17 (b)

2. State whether or not a triangle with sides of the given length is a right triangle.
a. 3m, 5m, 7m d. 10cm, 24 cm, 26 cm
b. 10m, 30m, 32m e. 20mm, 21mm, 29mm
c. 9cm, 12cm, 15cm f. 7km, 11km, 13km
3. Pythagorean triples consist of three whole numbers a, b and c which obey the
rule: a2 + b2 = c2
a. when a = 1 and b = 2, find the value of c.
b. when a = 3 and b = 4, find the value of c.
4. Pythagoras' Theorem states that a2 + b2 = c2 for the sides a, b and c of a right –
angled triangle. When a = 5, b = 12 then c = 13.
Find three more sets of rational numbers for a, b and c which satisfy
Pythagoras' Theorem.

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Theorem 7.3 (Pythagoras' Theorem)

If a right angled triangle has legs of lengths a and b and

hypotenuse of length c, then a2+b2=c2.
C

a
b

A B
c1 D c c2

Figure 7.18

Let ∆ ABC be right angled triangled the right angle at C as shown above:
Given: ∆ ACB is a right triangle and CD ⊥ AB.
We want to show that : a2+b2 = c2.
Proof:
Statements Reasons
2
1. a = c2  c 1. Euclid's Theorem
2. b2= c 1  c 2. Euclid's Theorem
3. a2+b2= (c 2 × c) + (c 1 × c) 3. Adding step 1 and 2
4. a2+b2=c(c 1 +c 2 ) 4. Taking c as a common factor
5. a2+b2=c(c) 6. Since c 1 +c 2 =c
6. a2+b2=c2 5. Proved
Example 4: If a right angle triangle ABC has legs of lengths a= 3cm and
b=4cm. What is the length of its hypotenuse?
Solution: Let c be the length of the hypotenuse
a2+b2=c2 . . . . Pythagoras' Theorem
(3cm)2+(4cm)2 = c2
9cm2+16cm2 = c2
25cm2 = c2
c = 25cm 2
c = 5cm
Therefore, the hypotenuse is 5 cm long.
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Example 5: If a right angle triangle ABC has leg of length a=24cm and the
hypotenuse c=25cm. Find the required leg.
Solution: If b is the length of the required leg, then
a2+b2 = c2 ........................ Pythagoras' Theorem
(24cm)2+b2 = (25cm)2
576cm2+b2 = 625cm2
b2 = (625-576)cm2
b2 = 49cm2
b= 49cm 2
b = 7cm
Therefore, the other leg is 7cm long.

The converse of the Pythagoras' Theorem is stated as follows:

Theorem 7.4 (Converse of Pythagoras' Theorem)

If the lengths of the sides of C

ABC are a, b and c where b a

a2+b2=c2 then the triangle is
right angled. The right angle is A c B
opposite the side of length c. Figure 7.19

C
The Pythagoras' Theorem and its
converse can be summarized as b a

follows respectively. A B
c
In ∆ABC with a, b lengths of Figure 7.20

the shorter sides and c the

Length of the longest side, then ∆ ABC is right angled if and
only if a2+b2=c2. Using the Figure above: ∆ ABC is right
angled, if and only if a2+b2=c2 .

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Example 6: In Figure 7.21 below. Is ∆ABC is a right-angled?
Solution: C
i.. a2= (12cm)2=144cm2
b = 5cm a = 12cm
ii. b2=(5cm)2=25cm2
ii. c2=(13cm)2=169cm2 A B
c = 13cm
Therefore, a2+b2=169cm2 and
Figure 7.21
c2=169cm2
Hence ∆ABC is right angled, the right angle at C….. converse of Pythagoras
theorem.

Example 7: In Figure 7.22 below. Is ∆ABC is a right-angled?

Solution:
A
i) a2= (8cm)2=64cm2
ii) b2= (12cm)2=144cm2
C = 15cm
iii) c2= (15cm)2= 225cm2 b = 12cm
Therefore, a2+b2 = (64+144) cm2
208cm2 and c2= 225 cm2
C B
Therefore, 208cm ≠ 225cm
2 2 a = 8cm
Figure 7.22
Therefore, ∆ ABC is not a right-angled.

Exercise 7B
1. In each of the following Figures ∆ABC is a right angled at C. Find the
unknown lengths of sides.
A
C A

6 cm x cm 5 cm y cm
10 cm

4 cm

B
A B C B C 24 cm
x cm 3 cm
(a) (b) (c)
Figure. 7. 23

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
2. In Figure 7.24 to the right, ABCD is a A B
rectangle with length and width 6 cm and
4 cm respectively. What is the length of the
diagonal AC?
D C
Figure 7.24

3. Find the height of an isosceles triangle with two congruent sides of length
37cm and the base of length 24cm.
4. Abebe and Almaz run 8km east and then 5km north. How far were they
from their starting point?
5. A mother Zebra leaves the rest of the herd to go in search of water. She
travels due south for 0.9km and, then due east for 1.2km. How far is she
from the rest of the herd?

Figure 7.25

6. In Figure 7.26 below ∆ABC is an equilateral triangle. AD ⊥ BC and

AB= 20 cm.
A
Find: a. AD b. BD c. DC
Hint: AD bisects both ∠BAC and BC.

B C
D

Figure 7.26

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
7. In Figure 7.27 to the right ABCD is a square A B
and DB the diagonal of the square BD= 6 2
cm. Find the length of side of the square.

D C
Figure 2.27

8. In Figure 7.28 to the right, Find the length of: A

3 cm D B
a. BE
b. DF 5 cm
E
c. EF F
8 cm
d. Is ∆CEF is a right-angled?
10 cm
e. Is ∆ADF is a right-angled?
C
Figure 2.28

9. The right-angled triangle ABC has sides 3cm, 4cm and 5cm. Squares have
been drawn on each of its sides. D

a. Find the number of cm squares in:

A E
i. the square CBFG I
ii. the square ACHI
H C B
iii. the square BADE
b. Add your answers for (a) (i) and (a) G F
(ii) above. Figure 7.29

c. State whether or not a triangle with sides of the given lengths is a right
triangle.

Using the theorems for calculations

Solving problems using the Euclid's and Pythagoras’ Theorems. You are now
well aware of the two theorems, their converses and their applications in
determining whether a given triangles is right angled triangle. You can now
summarize, the Euclid's and the Pythagoras’ theorems as follows:

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Given: a right angled triangle ABC as shown
C
in the figure to the right and CD is altitude to
the hypotenuse. Let a, b and c be the side b a

opposite to the angles A, B and C respectively.

If AD = c 1 and DB=c 2 , then A c1 D c2 B
c
a. a2 = c × c 2 Figure 7.30
………..Euclid's Theorem
b2 = c × c 1
b. a2+b2 = c2 ............. Pythagoras' Theorem
C
Example 8: In Figure 7.31 to the right, if
b a
DB = 8cm and AD = 4 cm
then find the lengths of:
A B
c1 D c2
c
a. AB b. BC c. AC d. DC Figure 7.31

Solution:
a. AB = AD + DB . . . . Definition of line segment.
AB = 4cm + 8cm
AB = 12cm
Hence c = 12cm
b. (BC)2 = (BD) × (BA) ...................... Euclid's Theorem
(BC)2 = (8cm) × (12cm)
(BC)2 = 96cm2
BC = 4 6 cm
c. (AC)2=(AD) × (AB) ...................... Euclid's Theorem
(AC)2=(4cm) × (12cm)
(AC)2 = 48cm2
AC = 48cm 2
AC = 4 3cm

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
2 2 2
d. (DC) + (BD) =(BC) ...................... Pythagorans' Theorem
(DC)2 + (8cm)2 = (4 6cm) 2
(DC)2 + 64cm2 = 96cm2
(DC)2 = (96-64) cm2
(DC)2 = 32 cm2
DC = 32cm 2
DC = 4 2cm
Example 9: In Figure 7.32 below, find the unknown (marked) length.
(AC)2 = (CD) × (CB) ....... Euclid's Theorem A
2
x = 1216
x
x2 = 192 unit square 8 h

x = 192 B C
4 D 12
x = 8 3 unit C
Figure 7.32
Therefore, the value of x= 8 3 unit.
(AD)2 + (DC)2 = (AC)2 ............................ Pythagoras' Theorem
h2 + (12)2 = (8 3 )2
h2 + 144 = 192
h2 = 192 − 144
h2 = 48
h = 48
h = 4 3 unit.

Exercise 7C A
4
1. In Figure 7.33, find x, a and b. D
a
x
8

B C
b
Figure 7.33
2. If p and q are positive integers such that p > q. Prove that p2-q2, 2pq and
p2+q2 can be taken as the lengths of the sides of a right-angled triangled.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
3. How long is an altitude of an equilateral Z
x
triangle if a side of the triangle is: W
a. 6cm long? b. a cm long? 5 6
h
4. In Figure 7.34 to the right, find x, y and h.
X Y
y
Figure 7.34
7.2 Introduction to Trigonometry
7.2.1 The Trigonometric Ratios

Activity 7.1
Discuss with your teacher
1. In Figure 7.35 below given a right angled triangle ABC
a. What is C
i. the opposite side to angle α?
β
ii. the adjacent side to angle α?
iii. the hypotenuse of ∆ ABC?
b. What is
i. the opposite side to β? α
A B
ii. the adjacent side to β? Figure 7. 35
iii. the hypotenuse of ∆ ABC?
2. In Figure 7.35 given a right angled triangle ABC:
a. In terms of the lengths AB, BC, AC, write sin α and sin β.
b. In terms of the lengths AB, BC, AC, write cos α and cos β.

The word trigonometric is derived from two Greek words trigono meaning a
triangle and metron meaning measurement. Then the word trigonometry literally
means the branch of mathematics which deals with the measurement of triangles.
The sine, the cosine and tangent are some of the trigonometric functions.
In this sub unit you are mainly dealing with trigonometric ratios. These are the
ratios of two sides of a right angled triangle.

? What are trigonometric ratios?

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Before defining them let us consider the
following Figure 7.36
Y
•
M3

M2

M1

α •
O
X1 X2 X3 X
Figure 7. 36

In Figure 7.36 OY and OX are rays that make an acute angle X1M1, X 2 M 2 and

X 3 M 3 are any three segments each from OY perpendicular to OX . It is obvious

to show that ∆OX 1 M 1 ~∆OX 2 M 2 ~∆OX 3 M 3 (by AA Similarity Theorem). Then
X1 M1 X 2 M 2 X 3 M 3
i. = = this ratio is called the sine of ∠XOY which is
OM 1 OM 2 OM 3

abbreviated as:
X1 M1 X 2 M 2 X 3 M 3
Sin (∠XOY)= = = = sin α, (sine ≅ sin).
OM 1 OM 2 OM 3
OX1 OX 2 OX 3
ii. = = this ratio is called the cosine of ∠XOY which is
OM 1 OM 2 OM 3

abbreviated as:
OX 1 OX 2 OX 3
Cos (∠XOY) = = = = cos α, (cosine ≅ cos).
OM 1 OM 2 OM 3

X1 M1 X 2 M 2 X 3 M 3
iii. = = this ratio is called the tangent of (∠XOY)
OX1 OX 2 OX 3

which is abbreviated as:

X1 M1 X 2 M 2 X 3 M 3
tan (∠XOY) = = = = tan α, (tangent ≅ tan).
OX1 OX 2 OX 3

Note: The sine, cosine and tangent are trigonometric ratio depends on
the measure of the angle but not on the size of the triangle.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
In Figure 7.37 to the right, in a right- A
triangle ABC, if ∠C is the right-angle,

adjacent
then AB is the hypotenuse, BC is the hypotenuse

side opposite to ∠A and AC is the side

adjacent to ∠A. B C
Opposite
Figure 7.37

Definition 7.1: If ∆ ABC is right-angled at C, then

length of the side opposite to ∠ A BC
a. sine ∠ A = =
length of hypotenuse AB

length of the side adjacent to < A AC

b. Cosine ∠ A = =
length of hypotenuse AB
length of the side opposite to ∠ A BC
c. Tangent ∠ A = =
length of the side adjacent to ∠A AC

Note: i. Sine of ∠ A, cosine of ∠ A and Tangent of ∠ A are respectively

abbreviated as sin ∠A, cos ∠A and tan ∠A.
ii. The lengths of the opposite side, adjacent side and hypotenuse
are denoted by the abbreviations "opp.", "adj." and "hyp."
Respectively.

Example 10: Use Figure 7.38 to state the value of each ratio.
B
a. sin A
7 x
b. cos A
c. tan A
A C
4
Figure 7.38

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Solution:
(AC)2 + (BC)2 = (AB)2 ................. Pythagoras’s Theorem
42 + x2 = 72
x2 = 49 − 16
x2 = 33
x= 33
opp. 33
a. Sin A = =
hyp. 7
adj. 4
b. Cos A = =
hyp. 7

opp. 33
c. tan A = =
adj.. 4

Exercise 7D
1. Use Figure 7.39 at the right to state the value of each ratio
a. sin θ A α B
θ
b. cos θ
c. tan θ 7
24
d. sin α
C
e. cos α
Figure 7.39
f. tan α
2. Use Figure 7.40 at the right to find
the value of each ratio. B

a. Find the value of x β

b. sin β
17
c. cos β x

d. tan β
e. sin θ θ
A C
f. cos θ 15
Figure 7. 40
g. tan θ
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Challenge Problems A

3. In Figure 7.41 at the right to state the β

value of each ratio.
z
a. sin β d. sin α y

b. cos β e. cos α
c. tan β f. tan α
α
C x B
Figure 7.41
4. Use Figure 7.42 at the right to describe each ratio.
sin β
a.
cos β C r
B
α
cos β
b.
sin β
S
sin α t
c.
cos α β

cos α Figure 7. 42
d. A
sin α

7.2.2 The Values of Sine, Cosine and tangent for 30°, 45° and 60°
The following class activity will help you to find the trigonometric values of the
special angle 45°.
Activity 7.2
Discuss with your friends/ parents
Consider the isosceles right angle triangle in Figure 7.43.
B
a. Calculate the length of the hypotenuse AB.
b. Are the measure angles A and B equal?
c. Which side is opposite to angle A? 1
d. Which side is adjacent to angle B?
e. What is the measure of angle A?
f. What is the measure of angle B? A
1 C
g. Find sin ∠A, cos ∠A and tan ∠A. Figure 7.43

h. Find sin ∠B, cos B and tan ∠ B.

i. Compare the result ( value) of g and h.

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From activity 7.3 you have found the values of sin 450 , cos 450 and tan 450. In an
isosceles right triangle, the two legs are equal in length. Also, the angles opposite
the legs are equal in measure. Since
B
0
m(∠A)+m(∠B)+m(∠c) = 180 and
45°
m(∠c) = 900 c= 2
1
m(∠A)+m(∠B) = 900
Since m(∠A)=m(∠B) each has the measure 450. 45°
A C
In Figure 7.44, each legs is 1 unit long. From the 1
Figure 7.44
Pythagorean property:
c2 = 12 + 12
c2 = 2
c= 2
opp. 1 2
Hence sin 450 = = = .... Why?
hyp. 2 2
adj. 1 2
cos 450= = = .... Why?
hyp. 2 2
opp.
tan 450= =1
adj.
Example 11: In Figure 7.45, find the values of x and y. B

Solution: y x
tan 45° = opp.
adj.
x 450
A
1= 3 C
3 Figure 7.45
x=3
sin 45° = opp.
adj.
1 3
=
2 y
y=3 2
The following Activity will help to find the trigonometric values of the special
angles 30° and 60°.
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Activity 7.3
Discuss with your friends/ partner
Consider the equilateral triangle ABC with side 2 units long as shown in Figure 7.46
below. C

a. Calculate the length of AD .

300 300
b. Calculate the length of DC . 2 2
c. Find sin 30°, cos 30° & tan 30°.
d. Find sin 60°, cos 60° and tan 60°.
600 600
e. Compare the results of sin 30° and cos 60°. A
D
B
f. Compare the results of cos 30° and sin 60°.
Figure 7.46
g. Compare the results of tan 30° and tan 60°.

C
From activity 7.3 you have attempted to find the
values of sin 300 , cos300, tan 300, sin 600,
300 300
cos600 and tan 600. Consider the equilateral 2 2
triangle in Figure 7.47 with side 2 units. The hh
altitude DC bisects ∠C as well as side AB. Hence
m(∠ACD)=300 and AD = 1 unit….. (Why)? 600 600
A B
D
2 2 2
(AD) + (DC) = (AC) ... Pythagorean Theorem Figure 7.47

in ∆ADC.
12 + h2 = 22
h2 + 1 = 4
h2 = 3
h = 3 units.
Now in the right-angled triangle ADC
C
opp. 1
Hence, sin 300= =
hyp. 2
300
adj. 3
cos 300= =
hyp. 2 2
opp. 1 3
tan 300= = =
adj. 3 3
opp. 3 A 600 D
sin 600= = 1
hyp. 2
Figure 7.48
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
adj. 1
cos 600= =
hyp. 2
opp. 3
tan 600= = = 3
adj. 1

Example 12: In Figure 7.49, find the values of x and y.

Solution:
opp. 5
sin 300= = x 5
hyp. x
1 5
= 300
2 x y
x=10 units Figure 7.49
0opp. 5
tan 30 = =
adj. y
3 5
=
3 y
3 y = 15
15
y= = 5 3 units
3
Example 13: In Figure 7.50, find the values B
of x and y. 600
4
Solution: x
opp. y
sin 600= =
hyp. 4 A C
y
3 y Figure 7.50
=
2 4
2y = 4 3
y = 2 3 unit.
adj. x
cos 600 = =
hyp. 4
1 x
=
2 4
x = 2 unit.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Example 14: A tree casts a 60 meter shadow and makes an angle of 300 with
the ground. How tall is the tree?
B

300
A
C
60m
Figure 7.51
Solution: Let Figure 7.51 represent the given problems.
opp.
tan ∠A=
adj.
h
tan 300=
60 m
h = 60m tan 30 0
3
h = 60m ×
3
h = 20 3 meter
Therefore, the height of the tree is 20 3 meters.
D C
Example 15: The diagonal of a rectangle is 20cm
long, and makes an angle of 300 20 cm
with one of the sides. Find the
lengths of the sides of the rectangle. A 300 B
Figure 7.52
Solution: Let Figure 7.52 represent the given problems
opp.
sin 300 =
hyp.
BC
sin 300=
20
BC = 20sin 300
1
BC = 20 ×
2
BC = 10cm

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
adj.
cos300 =
hyp.
AB
cos300 =
20 cm
3 AB
=
2 20 cm
AB = 10 3cm
Therefore, the lengths of the sides of the rectangles are 10cm and 10 3 cm.
Example 16: When the angle of elevation of the sun is 450, a building casts
a shadow 30m long. How high is the building?
Solution: Let Figure 7.53 represent sun

the given problem C

opp.
tan 450=
adj. h
h
1=
30 450
A
h = 30m 30m B

Therefore, the height of the Building is 30m. Figure 7.53

Example 17: A weather balloon ascends vertically at a rate of 3.86 km/hr while
it is moving diagonally at an angle of 600 with the ground. At the
end of an hour, how fast it moves horizontally (Refer to the
Figure 7.54 below).

Solution: Let Figure 7.54 represent the given

problem and x be the horizontal speed
opp.
tan 600 =
adj.
3.86 h=3.86 km/hr
3 = km/hr
x
3x = 3.86 km/hr
3.86 600
x= km/hr x
3
3.86 Figure 7.54
x= = 6.85km/hr
0.5774

Therefore, the horizontal speed of the ballon is 6.85km/hr.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Example 18: A ladder 20 meters long, leans against a wall and makes an angle
of 450 with the ground. How high up the wall does the ladder
reach? And how far from the wall is the foot of the ladder?
Solution: Let in Figure 7.55 represent the given problem
adj.
Cos 45° =
hyp. C
1 AB
= 20 m
2 20 m

WALL
20 = 2 AB
20 m
AB = = 10 2 meters 450
2 A
B Figure 7.55

Therefore, the foot of the ladder is 10 2 meters far from the wall.
opp.
sin 450 =
hyp.
1 BC
=
2 20 m
20 = 2 BC
20
BC = = 10 2 meters
2
Therefore, the ladder reaches at 10 2 meters
high far from the ground. CC
Example 19: At a point A, 30 meters from
the foot of a school building as shown
h
in Figure 7.56 to the right, the angle to
the top of the building C 600. What is 600
the height of the school, building? A 30 meters B
Solution: Figure 7.56
By considering ∆ ABC which is right angled you can
use trigonometric ratio.
opp.
tan 60 0 =
adj.
BC
tan 60 0 =
AB
h
3=
30m
h = 30 3meters
Therefore, the height of the school building is 30 3 meters.
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Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Exercise 7E
1. In Figure 7.57 below find the value of x.
C A

600

5 cm x

A 300 B B
C
x 8 cm
(a) (b)
Figure 7.57

2. A ladder of length 4m leans against a vertical wall so that the base of the
ladder is 2 meters from the wall. Calculate the angle between the ladder
and the wall.
3. A ladder of length 8m rests against a wall so that the angle between the
ladder and the wall is 450. How far is the base of the ladder from the wall?
4. In Figure 7.58 below, a guide wire is
used to support a 50 meters radio
antenna so that the angle of the wire
makes with the ground 600. How far
50 meters
is the wire is anchored from the base
of the antenna?
600
x
Figure 7.58

5. In an isosceles right triangle the length of a leg is 3cm. How long is the
hypotenuse?

Challenge problems
6. How long is an altitude of an equilateral triangles, if the length of a side
of the triangle is
a. 6cm b. 4cm c. 10 cm

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
0 0 0
7. In a 45 -45 -90 triangle the length of the hypotenuse is 20cm. How long
is its leg?

7.3 Solids Figures

7.3.1Pyramid

Historical note
Pyramids
The Egyptian pyramids are ancient pyramid
shaped brick work structures located in Egypt.
The shape of a pyramid is thought to be
representative of the descending rays of the
sun.

Group work 7.3

Discuss with your friends/ partners.
1. What is a pyramid?
2. Can you give a model or an example of a pyramid
3. Answer the following question based on the given Figure 7.59 below.
a. Name the vertex of the pyramid. V

b. Name the base of the pyramid.

c. Name the lateral faces of the pyramid.
d. Name the height of the pyramid.
e. Name the base edge of the pyramid. D E
C
f. Name the lateral edge of the pyramid.
A
B
Figure 7.59 Rectangular pyramid

Definition 7.2: A Pyramid is a solid figure that is formed by line segments

joining every point on the sides and every interior points of a
polygonal region with a point out side of the plane of the
polygon.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
From the group work (7.3) above you may discuss the following terminologies.
 The polygonal region ABCD is called the base of the pyramid.
 The point V outside of the plane of the polygon (base) is called the vertex
of the pyramid.
 The triangles VAB, VBC, VCD, and VDA are called lateral faces of the
pyramid (see Figure 7.59).
 AB, BC, CD and DA are the edges of the base of the pyramid (see Figure 7.59).
 VA, VB, VC and VD are lateral edge of the pyramid (see Figure 7.59).
 The altitude of a pyramid is the perpendicular distance from the vertex to
the point of the base.
 The slant height is the length of the altitude of a lateral face of the
pyramid.
 Generally look at Figure 7.60 below.

V Vertex
Lateral edge

Lateral face
height of the pyramid
D C Slant height

base
V' E
A
B
Figure 7.60 Rectangular pyramid

Figure 7.61 below show different pyramids. The shape of the base determines
the name of the pyramid.

V V
V

B
C B
B A C
A C
A D
D E
(a) Triangular pyramid (b) Quadrilateral pyramid (c) Pentagonal pyramid
Figure 7.61
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Activity 7.4
Discuss with your teacher before starting the lesson.
1. Make a list of the names of these shapes. You do not have to draw them.
Choose from: hexagonal pyramid, tetrahedron, and square pyramid.
2. What is a regular pyramid?
3. What is altitude of the pyramid?

a) b) c)
Figure 7.62
V
Special class of pyramids are known as
right pyramids. To have a right pyramid
the following condition must be satisfied:
The foot of the altitude must be at the
center of the base. In Figure 7.63 to the h
right shows a rectangular right pyramid.
The other class of right pyramids are D
known as regular pyramids. To have a C
regular pyramid, the following three A
B
conditions must be fulfilled:
Figure 7.63 Rectangular right pyramid
1. The pyramid must be a right pyramid.
2. The base of the pyramid must be a regular polygon.
In Figure 7.64 shows regular pyramids.
3. The lateral edges of a regular pyramid are all equal in length.

(a) Regular triangular pyramid (b) Regular square pyramid (c) Regular pentagonal
pyramid
Figure 7.64
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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Exercise 7F T

1. In Figure 7.65 shows a square pyramid.

a. Name its vertex.
b. Name its four lateral edges. 6 cm
Q R
c. Name its four lateral faces.
W 6 cm
d. Name the height.
P S
e. Name the base. 6 cm
Figure 7.65
V
2. In Figure 7.66 to the right
a. Name the vertex of the pyramid
b. Name the lateral edge of the pyramid
c. Name the lateral faces of the pyramid.
A C

B
Figure 7.66 Pyramid
7.3.2 Cone
Group Work 7.4
Discuss with your friends.
1. What is a cone? V
2. Answer the following question based on
the given Figure 7.67 to the right.

a. Name the vertex of the cone. h
b. Name the slant height of the cone.
c. Name the base of the cone.
d. Name the altitude of the cone. O E
Figure 7.67

Definition7.3: The solid figure formed by joining all points of a circle to a

point not on the plane of the circle is called cone.

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
In Figure 7.68, represent a cone.
Vertex
The original circle is called the base of V

the cone and the curved closed surface is Lateral face

its lateral surface.
Base
Plane
Figure 7.68

The point outside the plane and at which the segments from the circular region
joined is called the vertex of the cone.
The perpendicular distance from the base to the vertex is called the altitude of
the cone.

Definition7.4: a. A Right Circular cone is a V

Vertex
circular cone with the foot of
its altitude is at the center of
the base as shown in Figure Slant height

7.69 to the right.

Altitude
b. A line segment from the vertex
of a right circular cone to any Base
point of the circle is called the Figure 7.69
slant height.

Exercise 7G
1. Draw a cone and indicate:
a. the base d. the slant height
b. the lateral face e. the vertex
c. the altitude
2. What is right circular cone?
3. What is oblique circular cone?
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Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Summary For Unit 7

C
Given
b a

A B
c1 D c2
c

Figure 7.70

For 1-4 below refer to right triangle ABC in Figure 7.70 above.
1. Euclid's Theorem i) a2= c 2  c
ii) b2 = c1 × c
2. Converse of Euclid's Theorem.
a2= c 2 c and b2= c 1 c if and only if ∆ABC is right angled.
3. Pythagorean Theorem: a2+b2=c2. A
4. Converse of Pythagorean Theorem β
If a +b =c , then ∆ABC is right angled.
2 2 2
C
b hyp
5. Trigonometric ratio in right triangle opp
ABC where ∠C is the right angle
θ
(see Figure 7.71). C B
a adj
Figure 7.71
 is the side adjacent (adj) to angle θ.

 AC is the side opposite (opp) to angle θ.

 AB is the hypotenuse (hyp) to angle θ
opp. b opp. a
a. sin θ= = d. sin β = =
hyp. c hyp. c
adj. a adj. b
b. cos θ = = e. cosβ = =
hyp. c hyp. c
opp. b opp. a
c. tan θ= = f. tan β= =
adj. a adj. b

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Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
6. Referring to the values given in table 7.2 below.
θ Sinθ cosθ tanθ
300 1 3 3
2 2 3
450
2 2 1
2 2
600 3 1
2 3
2

7. Relationship between 300 and 600 as follows:

3
a. sin 600=cos 300=
2
1
b. cos 600=sin30 =
2
8. A Pyramid is a solid figure that is formed by line segments joining every point on
the sides and every interior points of a polygonal region with a point out side of the
plane of the polygon.
9. The solid figure formed by joining all points of a circular region to a point
outside of the plane of the circle is called a circular cone.
10. V
Vertex V Slant height
Lateral edge
h
Slant height
h
height Radius
D Lateral face
>E C r
base
A B
(a) Rectangular pyramid (b) Right circular cone
Figure 7.72

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]

Miscellaneous Exercise 7
I. Choose the correct answer from the given alternatives.
1. A rectangle has its sides 5cm and 12cm long. What is the length of its
diagonals?
a. 17cm b. 13 cm c. 7cm d. 12cm
2. In Figure 7.73 to the right C
m (∠ACB)=900 and CD ⊥ AB.
10cm
If CD = 10cm and BD = 8cm, then what
B A
is the length of AD ? 8cm D
Figure 7.73
a. 2 41 cm c. 41 cm
b. 4 41 cm d. cm
Y
3. In Figure 7.74 to the right, right angle
triangle XYZ is a right angled at Y 6cm
and N is the foot of the perpendicular
from Y to XZ . Given that XY= 6cm X Z
N
and XZ=10cm. What is the length of 10 cm
XN ? Figure 7.74

a. 2.4 cm c. 4.3 cm
b. 3.6 cm d. 4.8 cm
4. An electric pole casts a shadow of 24 meters long. If the tip of the shadow
is 25 meters far from the top of the pole, how high is the pole from the
ground?
a. 9 meters c. 7 meters
b. 10 meters d. 5 meters
5. Which of the following set of numbers could not be the length of sides of a
right angled triangle?
a. 0.75, 1, 1.25 c. 6, 8, 10
3
b. 1, , 2 d. 5, 12, 13
2
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Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
6. A tree 18 meters high is broken off 5 meters from the ground. How far
from the foot of the tree will the top strike the ground.
a. 12 meters b. 13 meters c. 8 meters d. 20 meters

13 m

5m

x
Figure 7.75

7. In Figure 7.76 below ∆ABC is right angled at C. if BC=5 and AB=13,

then which of the following is true?
A
12
a. sin θ =
13 θ
12
b. tan β = 13
5
5
c. cos θ =
13 β
B C
13
d. cos β = 5
5 Figure 7.76

8. In Figure 7.77 below, what is the value of x?

a. 6
b. 20
c. 10 x+4
x
d. 0

8
Figure 7.77

9. One leg of an isosceles right triangle is 3cm long. What is the length of
the hypotenuse?
a. 3cm b. 3 2 cm c. 3 3 cm d. 6 cm

216
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
10. In Figure 7.78 below which of the following is true?
C
16
a. sin ∠A=
22
16 16
b. cos ∠B= 2
22
57
c. tan ∠B = A B
8 22
d. All are correct answer Figure 7.78

11. In Figure 7.79 below, which of the following is true about the value of
the variables? 2 B
a. x=2 3 D
x 4
b. y=6 y

c. z=4 3 A C
z
d. All are true Figure 7.79

12. In Figure 7.80 to the right What is the value of x?

a. 6 A
b. 18
c. 3 2 6

d. -3 2

B C
x
Figure 7.80
13. In Figure 7.81 below, what is the value of x?
a. 10 3 A
30
b.
3
10
30 3
c.
3
300
d. All are true B x C

Figure 7.81

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 Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
14. Which of the following is true
Q
about given ∆PQR given in
Figure 7.82 to the right? r
a. p2+q2=r2 P
2 2 2
b. q +r =p
c. (p+q)2=r2 R P
q
d. p2+r2=q2
Figure 7.82

15. In Figure 7.83 to the right, find the B

length of the side of a rhombus whose
diagonals are of length 6 and 8 unit.
8
a. 14 units A 6 C
b. 5 units
c. 10 unit
d. 15 unit D
Figure 7.83
II. Work out Question
C
16. In Figure 7.84 to the right, CD ⊥ AB
AD = 4 cm, CD = 9cm and DB=14cm.
Is ∆ABC a right angled? 9 cm
A B
D
4 cm 14 cm
Figure 7.84

B
17. ∆ABC is a right-angled triangle as
shown in Figure 7.85 to the right. If
AD = 12cm BD = DC then find the
lengths of BD and DC . A C
12cm
D
Figure 7.85

218
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
18. In Figure 7.86 to the right, find the
A
value of the variables.
3
D
19. A triangle has sides of lengths 16, y
x
48 and 50. Is the triangle a right-
angled triangle? B C
2x
A Figure 7.86
20. In Figure 7.87 to the right, if
AC = 12 cm, BC = 5 cm, x
CD = 11 cm, then find y 12 cm
a. b.

B 5 cm C 11 cm D
Figure 7.87
21. Let ∆ABC be an isosceles triangle
and AD be its altitude. If the length
A
of side AC = 4x + 4y, BD = 6x,
DC = 2x + 2y and AB = 12, then
find the length of:
h
a. AC d. BD
b. AD e. DC
B C
c. BC D
Figure 7.88

22. In Figure 7.89 to the right, what is C

2
the value of x, if sin B = . x + 45
3 45 - x

A B
Figure 7.89

219
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
23. In Figure 7.90 to the right, what is F
8
the value of x, if tan∠ D = .
5
13 + x

E D
13 - x
Figure 7.90
24. In Figure 7.91 to the right, what is the
A
2
value of x, if cos c= .
5
x + 21

B C
x

Figure 7.91

25. In Figure 7.92 below, if BC = 5 and AB = 13, then find

a. sinα
b. cos α A

c. tanα α
d. sinβ 13
e. cosβ
f. tanβ
sin α sin β β
g. + B
cos α cos β 5 C
h. 2
(sinα) +(cosα) 2 Figure 7.92

i. (cosβ)2+(sinβ )2

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Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
The Function y = x2 1.00 < x < 5.99
X 0 1 2 3 4 5 6 7 8 9
1.0 1.000 1.020 1.040 1.061 1.082 1.102 1.124 1.145 1.166 1.188
1.1 1.210 1.232 1.254 1.277 1.277 1.322 1.346 1.369 1.392 1.416
1.2 1.440 1.464 1.488 1.513 1.513 1.562 1.588 1.613 1.638 1.644
1.3 1.690 1.716 1.742 1.769 1.769 1.822 1850 1.877 1.904 1.932
1.4 1.960 1.988 2.016 2.045 2.045 2.102 2.132 2.161 2.190 2.220
1.5 2.250 2.280 2316 2.341 2.341 2.102 2.434 2.465 2.496 2.528
1.6 2.560 2.592 2.310 2.657 2.657 2.722 2.756 2.789 2.822 2.856
1.7 2.890 2.924 2.624 2.993 2.993 3.062 3.098 3.133 3.168 3.204
1.8 3.240 3.276 2.958 3.349 3.349 3.422 3.460 3.497 3.534 3.572
1.9 3.610 3.648 3.686 3.725 3.764 3.802 3.842 3.881 3.920 3.960
2.0 4.000 4.040 4.080 4.121 4.162 4.202 4.244 4.285 4.326 4.368
2.1 4.410 4.452 4.494 4.537 4.580 4.622 4.666 4.709 4.752 4.796
2.2 4.840 4.884 4.928 4.973 5.018 5.062 5.108 5.153 5.198 5.244
2.3 5.290 5.336 5.382 5.429 5.476 5.522 5.570 5.617 5.664 5.712
2.4 5.760 5.808 5.856 5.905 5.954 6.002 6.052 6.101 6.150 6.200
2.5 6.250 6.300 6.350 6.401 6.452 6.502 6.554 6.605 6.656 6.708
2.6 6.760 6.812 6.864 6.917 6.970 7.022 7.076 7.129 7.182 7.236
2.7 7.290 7.344 7.398 7.453 7.508 7.562 7.618 7.673 7.728 7.784
2.8 7.840 7.896 7.952 8.009 8.066 8.122 8.180 8.237 8.294 8.352
2.9 8.410 8.468 8.526 8.585 8.644 8.702 8.762 8.821 8.880 8.940
3.0 9.000 9.060 9.120 9.181 9.242 9.302 9.364 9.425 9.486 9.548
3.1 9.610 9.672 9.734 9.797 9.860 9.922 9.986 10.05 10.11 10.18
3.2 10.24 10.30 10.37 10.43 10.50 10.56 10.63 10.69 10.76 11.82
3.3 10.89 10.96 11.02 11.09 11.16 11.22 11.29 11.36 11.42 11.49
3.4 11.56 11.63 11.70 11.76 11.83 11.90 11.97 12.04 12.11 12.18
3.5 14.25 12.32 12.39 12.46 12.53 12.60 12.67 12.74 12.82 12.89
3.6 12.96 13.03 13.10 13.18 13.25 13.32 13.40 13.47 13.54 13.62
3.7 13.69 13.76 13.84 13.91 13.99 14.06 14.14 14.21 14.29 14.36
3.8 14.44 14.52 14.59 14.67 14.75 14.82 14.90 14.98 15.08 15.13
3.9 15.21 15.29 15.37 15.44 15.52 15.60 15.68 15.76 15.84 15.92
4.0 16.00 16.08 16.16 16.24 16.32 16.40 16.48 16.56 16.65 16.73
4.1 16.81 16.89 16.97 17.06 17.14 17.22 17.31 17.39 17.47 17.56
4.2 17.64 17.72 17.81 17.89 17.98 18.06 18.15 18.23 18.32 18.40
4.3 18.49 18.58 18.66 18.75 18.84 18.92 19.01 19.10 19.18 19.27
4.4 19.96 19.45 19.54 19.62 19.71 19.80 19.89 19.98 20.98 20.16
4.5 20.25 20.34 20.43 20.52 20.61 20.70 20.79 20.88 21.90 21.07
4.6 21.16 21.25 21.34 21.44 21.53 21.62 21.72 21.81 22.85 22.00
4.7 22.09 22.18 22.28 22.37 22.47 22.56 22.66 22.75 23.81 22.94
4.8 23.04 23.14 23.33 23.33 23.43 23.52 23.62 23.72 24.80 23.91
4.9 24.01 24.11 24.24 24.30 24.40 24.50 24.60 24.70 25.81 24.90
5.0 25.00 25.10 25.20 25.30 25.40 25.50 25.60 25.70 26.83 25.91
5.1 26.01 26.11 26.21 26.32 26.42 26.52 26.63 26.73 27.88 26.94
5.2 27.04 27.14 27.25 27.35 27.46 27.56 27.67 27.77 28.94 27.98
5.3 28.09 28.20 28.30 28.41 28.52 28.62 28.73 28.84 28.94 29.05
5.4 29.16 29.27 29.38 29.48 29.59 29.70 29.81 29.92 30.03 30.14
5.5 30.25 30.36 30.47 30.58 30.69 30.80 30.91 31.02 31.14 31.25
5.6 31.36 31.47 31.58 31.70 31.81 31.92 32.04 32.15 32.26 32.38
5.7 32.46 32.60 32.72 32.83 32.95 33.06 33.18 33.29 33.41 33.52
5.8 33.64 33.76 33.87 33.99 34.11 34.22 34.34 34.46 34.57 34.69
5.9 34.81 34.93 35.05 35.16 35.28 35.40 35.52 35.64 35.76 35.88

If you move the comma in x one digit to the right (left), then the comma in x2 must be
moved two digits to the right (left)

221
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
The Function y = x2 6.00 < x < 9.99
X 0 1 2 3 4 5 6 7 8 9
6.0 36.00 36.12 36.24 36.36 36.48 36.60 36.72 36.84 36.97 37.09
6.1 37.21 37.33 37.45 37.58 37.70 37.82 37.95 38.07 38.19 38.32
6.2 38.44 38.56 38.69 38.81 38.94 39.06 39.19 39.31 39.44 39.56
6.3 39.69 39.82 39.94 40.07 40.20 40.32 40.45 40.58 40.70 40.83
6.4 40.96 41.09 41.22 41.34 41.47 41.60 41.73 41.86 41.99 42.12
6.5 42.25 42.38 42.51 42.64 42.77 42.90 43.03 43.16 43.30 43.43
6.6 43.56 43.69 43.82 43.96 44.09 44.22 44.36 44.49 44.62 44.47
6.7 44.89 45.02 45.16 45.29 45.43 45.56 45.70 45.63 45.63 46.10
6.8 46.24 46.38 46.51 46.65 46.79 46.92 47.06 47.20 47.20 47.47
6.9 47.61 47.75 47.89 48.02 48.16 48.30 48.44 48.58 48.58 48.86
7.0 49.00 49.14 49.28 49.42 49.56 49.70 49.84 49.98 49.98 50.27
7.1 50.41 50.55 50.69 50.84 50.98 51.12 51.27 51.41 54.41 51.70
7.2 51.84 51.98 52.13 52.27 52.42 52.56 52.71 52.85 52.85 53.14
7.3 53.29 53.44 53.58 53.73 53.88 54.02 54.17 54.32 54.32 54.61
7.4 54.76 54.91 55.06 55.20 55.35 55.50 55.65 55.80 55.80 56.10
7.5 56.25 56.40 56.55 56.70 56.85 57.00 57.15 57.30 57.30 57.61
7.6 57.76 57.91 58.06 58.22 58.37 58.52 58.68 58.83 58.83 59.14
7.7 59.29 59.44 59.60 59.75 59.91 60.06 60.22 60.37 60.37 6.68
7.8 60.84 61.00 61.15 61.31 61.47 61.62 61.78 61.94 61.94 62.25
7.9 62.41 62.57 62.73 62.88 63.04 63.20 63.36 63.52 63.52 63.84
8.0 64.00 64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65.45
8.1 65.61 65.77 65.93 66.10 66.26 66.42 66.59 66.75 66.91 67.08
8.2 67.24 67.40 67.57 67.73 67.90 68.06 68.23 68.39 68.56 68.72
8.3 68.29 69.06 69.22 69.39 69.56 69.72 69.89 70.06 70.22 70.39
8.4 70.56 70.73 70.90 71.06 71.23 71.40 71.57 71.74 71.91 72.01
8.5 72.25 72.42 72.59 72.76 72.93 73.10 73.27 73.44 73.62 73.79
8.6 73.96 74.13 74.30 74.48 74.65 74.82 75.00 75.17 75.34 75.52
8.7 75.69 75.86 76.04 76.21 76.39 76.56 76.74 76.91 76.09 77.26
8.8 77.44 77.62 77.79 77.97 78.15 78.32 78.50 78.68 78.85 79.03
8.9 79.21 79.39 79.57 79.74 79.92 80.10 80.28 80.46 80.46 80.82
9.0 81.00 81.18 84.36 81.54 81.72 81.90 82.08 82.26 82.45 82.63
9.1 82.81 82.99 83.17 83.36 83.54 83.72 83.91 84.09 86.12 84.46
9.2 84.64 84.82 85.01 85.19 85.38 85.56 85.75 84.93 87.96 86.30
9.3 86.49 86.68 86.86 87.05 87.24 87.42 87.61 87.80 87.96 88.17
9.4 88.36 88.55 88.74 88.92 89.11 89.30 89.49 89.68 89.87 90.06
9.5 90.25 90.44 90.63 90.82 91.01 91.20 91.39 91.58 91.78 91.97
9.6 92.16 92.35 92.54 92.74 92.93 93.12 93.32 93.51 93.70 93.90
9.7 94.09 94.28 94.67 94.67 94.87 95.06 95.26 95.45 95.65 95.84
9.8 96.04 96.24 96.43 96.63 96.83 97.02 97.22 97.42 97.61 97.81
9.9 98.01 98.21 98.41 98.60 98.80 99.00 99.20 99.40 99.60 99.80

(8.47)2= 71.74 (0.847) 2= 0.7174 = 0.463

(84.7)2 = 7174 (8.472)2 = 71.77

222
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Y = x3 1.00 < x < 5.99
X 0 1 2 3 4 5 6 7 8 9
1.0 1.000 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295
1.1 1.331 1.368 1.405 1.443 1.482 1.521 1.561 1.602 1.643 1.685
1.2 1.728 1772 1.816 1.861 1.907 1.953 2.000 2.048 2.097 2.147
1.3 2.197 2.248 2.300 2.353 2.406 2.460 2.515 2.571 2.628 2.686
1.4 2.744 2.803 2.863 2.924 2.986 3.049 3.112 3.177 3.242 3.308
1.5 3.375 3.443 3.512 3.582 3.652 3724 3.796 3.870 3.944 4.020
1.6 4.096 4.173 4.252 4.331 4.411 4.492 4.574 4.657 4.742 4.827
1.7 4.913 5.000 5.088 5.178 5.268 5.359 5.452 5.545 5.640 5.735
1.8 5.832 8.930 6.029 6.128 6.230 6.332 6.435 6.539 6.645 6.751
1.9 6.859 6.968 7.078 7.189 7.301 7.415 7.530 7.645 7.762 7.881
2.0 8.000 8.121 8.242 8.365 8.490 8.615 8.742 9.870 8.999 9.129
2.1 9.261 93.394 9.528 9.664 9.800 9.938 10.08 10.22 10.36 10.50
2.2 10.65 10.79 11.94 11.09 11.24 11.39 11.54 11.70 11.85 12.01
2.3 12.17 12.33 12.49 12.65 12.81 12.98 13.14 13.31 31.48 13.65
2.4 13.82 14.00 14.17 14.35 14.53 14.71 14.89 15.07 15.25 15.44
2.5 15.63 15.81 16.00 16.19 16.39 16.58 16.78 16.97 17.17 17.37
2.6 17.58 17.78 17.98 18.19 18.40 18.61 18.82 19.03 19.25 19.47
2.7 19.68 19.90 20.12 20.35 20.57 20.08 21.02 21.25 21.48 21.72
2.8 21.95 22.19 22.43 22.67 22.91 23.15 23.39 23.64 23.89 24.14
2.9 24.39 24.64 24.90 25.15 25.41 25.67 25.93 26.20 25.46 26.73
3.0 27.00 27.27 27.54 27.82 28.09 28.37 28.65 28.93 29.22 29.50
3.1 29.79 30.08 30.37 30.66 30.96 26 31.55 31.86 32.16 32.46
3.2 32.77 33.08 33.39 33.70 34.01 34.33 34.65 34.97 35.29 35.61
3.3 35.94 36.26 36.59 36.93 37.26 37.60 37.93 38.27 38.61 38.96
3.4 39.30 39.65 40.00 40.35 40.71 41.06 41.42 41.78 42.14 42.51
3.5 42.88 43.24 43.61 43.99 44.36 44.47 45.12 45.50 45.88 46.27
3.6 46.66 47.05 47.44 47.83 48.23 48.63 49.03 49.43 49.84 50.24
3.7 50.65 51.06 51.48 51.90 52.31 52.73 53.16 53.58 54.01 54.44
3.8 54.87 55.31 55.74 56.18 56.62 57.07 57.51 57.96 58.41 58.86
3.9 59.32 59.78 60.24 60.70 61.16 61.63 62.10 62.57 63.04 63.52
4.0 64.00 64.48 64.96 65.45 65.94 66.43 66.92 67.42 67.92 68.42
4.1 68.92 69.43 69.913 70.44 70.96 71.47 71.99 72.51 73.03 73.56
4.2 74.09 74.62 75.15 75.69 76.23 76.77 77.31 77.85 78.40 79.95
4.3 79.51 80.06 80.62 81.18 81.75 82.31 82.88 83.45 84.03 84.60
4.4 85.18 85.77 86.35 86.94 87.53 88.12 88.72 89.31 89.92 90.52
4.5 91.13 91.73 92.35 92.96 93.58 94.20 94.82 95.44 96.07 96.70
4.6 97.34 97.97 98.61 99.25 99.90 100.5 101.2 101.8 102.5 103.2
4.7 103.8 104.5 105.2 105.8 106.5 107.2 107.9 108.5 109.2 109.9
4.8 110.6 111.3 112.0 112.7 113.4 114.1 114.8 115.5 116.2 116.9
4.9 117.6 118.4 119.1 119.8 120.6 121.3 122.0 122.8 123.5 124.3
5.0 125.0 125.8 126.5 127.3 128.0 128.8 129.6 130.3 131.1 131.9
5.1 132.7 133.4 134.2 135.0 135.8 136.6 137.4 138.2 139.0 139.8
5.2 140.6 141.4 142.2 143.1 143.9 144.7 145.5 146.4 147.2 148.0
5.3 148.9 149.7 150.6 151.4 152.3 153.1 154.0 154.5 155.7 156.6
5.4 157.5 158.3 159.2 160.1 161.0 161.9 162.8 163.7 164.6 165.5
5.5 166.4 167.3 168.2 169.1 170.0 171.0 171.9 172.8 173.7 174.7
5.6 175.6 176.6 177.5 178.5 179.4 180.4 181.3 182.3 183.3 184.2
5.7 185.2 186.2 187.1 188.1 189.1 190.1 191.1 192.1 193.1 194.1
5.8 195.1 196.1 197.1 198.2 199.2 200.2 201.2 202.3 203.3 204.3
5.9 205.4 206.4 207.5 208.5 209.6 210.6 211.7 212.8 213.8 214.9

If you move the comm. In x one digit to the right (left), then the comma in x3 must be
moved three digits to the right (left)

223
Grade 8 Mathematics [GEOMETRY AND MEASUREMENT]
Y = x3 6.00 < x < 9.99
X 0 1 2 3 4 5 6 7 8 9
6.0 216.0 217.1 218.2 219.3 220.3 221.4 211.7 223.6 224.8 225.9
6.1 227.0 228.1 229.2 230.0 231.5 232.6 222.5 234.9 236.0 237.2
6.2 238.3 239.5 240.6 241.8 243.0 244.1 233.7 246.5 247.7 248.9
6.3 250.0 251.2 252.4 253.6 254.8 256.0 245.3 258.5 259.7 260.9
6.4 262.1 263.4 264.6 265.8 267.1 268.3 257.3 270.8 272.1 273.4
6.5 274.6 275.9 277.2 278.4 279.7 281.0 269.6 283.6 284.9 286.2
6.6 287.5 288.8 290.1 291.4 292.8 294.1 282.3 296.7 298.1 299.4
6.7 300.8 302.1 303.5 304.8 306.2 307.5 295.4 310.3 311.7 313.0
6.8 314.4 315.8 317.2 318.6 320.0 321.4 308.9 324.2 325.7 327.1
6.9 328.5 329.9 331.4 332.8 334.3 335.7 322.8 324.2 340.1 341.5
7.0 343.0 344.5 345.9 347.4 348.9 350.4 337..2 338.6 354.9 356.4
7.1 357.9 359.4 360.9 362.5 364.0 365.5 351.9 368.6 370.1 371.7
7.2 373.2 374.8 376.4 377.9 379.5 381.1 367.1 384.2 385.8 387.4
7.3 389.0 390.6 392.2 393.8 395.4 397.1 382.7 400.3 401.9 403.6
7.4 405.2 406.9 408.5 410.2 411.5 413.5 398.7 416.8 418.5 420.2
7.5 421.9 423.6 425.3 427.0 428.7 430.4 415.2 433.8 435.5 437.2
7.6 439.0 440.7 442.5 444.2 445.9 447.7 432.1 451.2 453.0 454.8
7.7 456.5 458.3 460.1 461.9 463.7 465.5 449.5 469.1 470.9 472.7
7.8 474.6 476.4 478.2 480.0 481.9 483.7 467.3 487.4 489.3 491.2
7.9 493.0 494.9 496.8 498.7 500.6 502.5 485.6 506.3 508.2 510.1
8.0 512.0 513.9 515.8 514.8 519.7 521.7 504.4 525.6 527.5 529.5
8.1 531.4 533.4 535.4 537.4 539.4 541.3 523.6 545.3 547.3 549.4
8.2 551.4 533.4 555.4 557.4 559.5 561.5 543.3 565.6 567.7 569.7
8.3 571.8 573.9 575.9 578.0 580.1 582.2 563.6 586.4 588.5 590.6
8.4 592.7 594.8 596.9 599.1 601.2 603.4 584.3 607.6 609.8 612.0
8.5 614.1 616.3 618.5 620.7 622.8 625.0 605.5 629.4 631.6 633.8
8.6 636.1 638.3 640.5 642.7 645.0 647.2 627.2 651.7 654.0 656.2
8.7 658.5 660.8 663.1 665.3 667.6 669.9 649.5 674.5 676.8 679.2
8.8 681.5 683.8 686.1 688.5 690.8 693.2 672.2 697.9 700.2 702.6
8.9 705.0 707.3 709.7 712.1 714.5 716.9 695.5 721.7 724.2 726.6
9.0 729.0 731.4 733.9 736.3 738.8 741.2 719.3 746.1 748.6 751.1
9.1 753.6 756.1 758.6 761.0 763.6 766.1 743.7 771.1 773.6 776.2
9.2 778.7 781.2 783.8 786.3 788.9 791.5 768.6 796.6 799.2 801.8
9.3 804.4 807.0 809.6 812.2 814.8 817.4 794.0 822.7 825.3 827.9
9.4 830.6 833.2 835.9 838.6 814.2 843.9 820.0 849.3 852.0 854.7
9.5 857.4 860.1 862.8 865.5 868.3 871.0 846.6 876.5 879.2 882.0
9.6 884.7 887.5 8890.3 893.1 895.8 898.6 873.7 904.2 907.0 909.9
9.7 912.7 915.5 918.3 921.2 924.0 926.9 901.4 932.6 935.4 938.3
9.8 941.2 944.1 947.0 449.0 952.8 955.7 929.7 961.5 964.4 967.4
9.9 970.3 973.2 976.2 979. 982.1 985.1 958.6 991.0 994.0 997.0

(8.47)3 = 607.6 (0.847)3 = 0.607 3

123.5 = 4.98 3 0.1235 = 0.498

(84.7)3 = 607600 (8.472)3 = 608.0 3 123500 = 49.8

224