ACKNOWLEDGEMENT

The completion of this report on Analytical Geometry has been a rewarding journey made possible
through the combined efforts of an exceptional team. I am truly grateful for the unwavering
commitment and enthusiasm shown by every group member throughout the course of this project.

This report is the outcome of active collaboration and mutual support. Sincere appreciation is
extended to all team members, Ashesh Kunwar, Ashok Khanal, Asmita Gyawali, Ayush
Lamichhane, Ayush Rajak, Bidhan Khanal, Aryan Man Shrestha, Asim Subedi, Ashutosh RD,
Bivek Upadhyaya. Our valuable ideas, continuous participation, and dedication were instrumental in
shaping the content and direction of our work.

We consider it a privilege to have worked alongside each other. Together, we not only deepened our
understanding of the subject but also strengthened our ability to work as a unified team. This report is
a reflection of our shared knowledge, effort, and a strong sense of teamwork.

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 Table of Contents
• Acknowledgement ................................................................................................................. 1

• Introduction ........................................................................................................................... 3
Definition…………………………………………………………………………………………….4

• Equation of a Straight Line and its forms .......................................................................... 4

• Applications of the Standard Forms of Equation of Straight Lines ................................ 6

• Derivation of Perpendicular Form ...................................................................................... 7

Point of intersection…………………………………………………………………………………7

Equation of a line passing through intersection……………………………………………………..8

• Concurrent Lines .................................................................................................................. 9

Definition……………………………………………………………………………………………9

Condition of concurrency……………………………………………..…………………………… 9

• Angle Between Lines ........................................................................................................... 10

• Length of Perpendicular ..................................................................................................... 11

Remarks…………………………………………………………………………………………….12

• Two Sides of a Line ............................................................................................................. 12

• Equations of Bisectors ........................................................................................................ 14

• Pair of Straight Lines ......................................................................................................... 15

Homogeneous equations.....................................................................................................................15

• Condition for a Second Degree Equation to Represent a Pair of Straight Lines.……...16

• Conclusion ........................................................................................................................... 19

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Introduction

Analytic geometry also known as coordinate geometry was initiated by French philosopher Rene
Descartes in his book, La Geometric (1637) where he introduced algebra in the study of points
andlocus of points. Thus, the analytic or coordinate geometry provides a connection between algebra
and geometry through the graphs of curves and lines. With the help of coordinate geometry, we can
solve geometric problems algebraically. Moreover, it is also enables as to get geometric insights into
algebra, Analytic geometry is widely needed in various situations in calculus and trigonometry.
Therefore, it is considered as one of the mostly needed concepts of mathematics. Coordinate
Geometry is needed in many real life causes such as in digital world, in describing position of any air
traffic control), in map projections, in the problems of latitude and longitude, etc.
Analytic Geometry is a branch of algebra, a great invention of Descartes and Fermat, which deals
with the modelling of some geometrical objects, such as lines, points, curves, and so on. It is a
mathematical subject that uses algebraic symbolism and methods to solve the problems. It establishes
the correspondence between the algebraic equations and the geometric curves. The alternate term
which is used to represent the analytic geometry is “Coordinate Geometry”.
It covers some important topics such as midpoints and distance, parallel and perpendicular lines on
the coordinate plane, dividing line segments, distance between the line and a point, and so on. The
study of analytic geometry is important as it gives the knowledge for the next level of mathematics. It
is the traditional way of learning the logical thinking and the problem solving skills. In this article, let
us discuss the terms used in the analytic geometry, formulas, Cartesian plane, analytic geometry in
three dimensions, its applications, and some solved problems.

In this chapter, we focus on fundamental geometric objects namely straight line and pair of straight
lines. We study these objects in detail and derive various relations.

Objectives of analytic geometry

 It establishes a correspondence between geometric curves and algebraic equations.
 It helps provide a visual representation of geometric problems
 It aids in utilizing co-ordinates to understand 2/3 dimensional objects.
 It lets computer graphics programs efficiently change the shape or view of pictured objects.

Material Collection
All the contents of this project work were written by evaluating information regarding the topic from
various different sources. The contents of course books, external books, external websites regarding
‘Analytical Geometry’ were carefully evaluated by the group members. They were then summarized
and then listed down as given below.

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What is a straight line?

A straight line, often known as a line, is a one-dimensional infinite geometry having no width but just
length. There are an endless number of points in a straight line. A two-variable linear equation is used
to represent a straight line. A pair of straight lines is represented by a second-degree equation in two
variables under specific conditions. When the product of two linear equations in x and y indicates a
straight line is multiplied together, a pair of straight lines is generated. ax2+2hxy+by2=0 is a second-
degree homogeneous equation that depicts a pair of straight lines flowing through the origin.If
tanθ=0, two lines will be parallel or coincident. i.e. if h2–ab=0.If tanθ is not defined, a+b=0, two
lines will be perpendicular.
Equation of a straight line

REDUCTION OF ax + by + C = 0 INTO THREE STANDARD FORMS

The three standard forms of the equations are:
1. Slope form

In mathematical terms, the slope is the rate of change of y with respect to x. When dealing
with linear equations, we can easily identify the slope of the line represented by the equation
by putting the equation in slope-intercept form, y = mx + b, where m is the slope and b is the
y-intercept.

2. Intercept form

The intercept form of the equation of a line is x/a + y/b = 1. This is one of the important forms
of equations of a line. Also, the sign of the intercepts in this equation helps us to know the
location of the line with respect to the coordinate axes. The intercept form of the equation of
the line can be understood as the line which makes a right triangle with the coordinates axes,
with the sides of lengths as 'a' units and 'b' units respectively.
The x-intercept is the shortest distance of the point on the x-axis from the origin, where the
line cuts the x-axis, and the y-intercept is the shortest distance of the point on the y-axis from
the origin, where the line cuts the y-axis.

3. Perpendicular form

The perpendicular form of a line's equation is given by x cos(α) + y sin(α) = p, where p is the
length of the perpendicular from the origin to the line, and α is the angle the perpendicular
makes with the positive x-axis. This form is also known as the normal form.

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To reduce ax+by+c=0 in slope intercept form y = mx+c.
We have,
ax+by+c=0 ……. (1)
or, by = -ax – c
a c
or, y = - x - …….. (2)
b b

is in the form of y = mx + C,
−a
where, m =
b
−c
C= .
b

Thus, (1) is in the form y = mx +C.

x y
To reduce ax + by +c =0 in the double intercept form + = 1.
A B

Dividing both sides of (1) by C, C ≠ 0, we get

ax by
+ =1
−c −c

x y
x y
or, −c + −c = 1, is in the form + = 1.
A B
a b
c c
Where, x intercept A= - and y intercept B = - .
a a

To reduce ax+by+c = 0 in the normal form , that is , in the form

xcos α + ysin α = p ……………(1)
And ax+by+c=0 …………..(2)
Equation (1) and (2) are identical
cos α sin α −p
∴ = = = k (say)
a b c
→ cos α = ak …………..(3)
sin α = bk ………….(4)
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From (3) and (4), squaring and adding both sides
cos α + sin α = k2 (a2+b2)
2 2

1
∴k=±
√ a +b2
2

From (5), to make p positive, we have to choose c and k of opposite signs.

a b c
∴ cos α = - , sin α = - ,p= , c ¿0 , and k ¿0.
√ a +b
2 2
√ a +b
2 2
√ a +b2
2

So, equation (1) takes the form.

ax by −c
- - = ,
√ a +b
2 2
√ a +b
2 2
√ a2 +b2
Which is the required equation in normal form .

ax by −c
For c¿0 , + = .
√ a +b
2 2
√ a +b
2 2
√ a2 +b2

Applications of the standard forms:

For Slope form:

1. Price Determination
Used to model how 1. Economics – Price Determination product prices change with demand.
Example: P=3D+9P = 3D + 9P=3D+9 shows price increases by $3 per unit demand.

2 Structural Design:
Used to model changes in structure dimensions.
Example: y=−0.1x+50y = -0.1x + 50y=−0.1x+50 models how bridge height changes from center to
ends.

3. Finance – Investment Analysis

Used in regression analysis to relate stock returns to market performance.
Example: y=0.8x+0.5y = 0.8x + 0.5y=0.8x+0.5, where slope (Beta) indicates risk level.

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Derivation Of Perpendicular form of Equation.

The equation of a straight line when the length of the perpendicular form the origin on the line and
the angle made by this perpendicular with the positive direction of the x- axis are given.
Let AB be the required line and p be the perpendicular form (0,0) on AB , so that OP = p . Let α be
the angle between OP and the positive direction of x-axis.

Y B

P
p
90° - α
α

O A X

Fig. 1
OP p
cos α = =
OA OA
p
or , OA =
cos α
OP
And cos (90° -α ) =
OB

We know from double intercept from

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 x y
+ =1
OA OB
x y
+
or, p p =1
cos α sin α

or, xcos α + y sin α = p

∴ x cos α + y sin α -p = 0, is the equation of the straight line AB.

The Point of Intersection of Two Lines

Let the two straight lines whose equations are:

a1x + b1y + c1 = 0 …(1)
and, a2x + b2y + c2 = 0 …(2)
intersect at the point P(x1,y1)
Therefore
a1x1 + b1y1 + c1 = 0 …(3)
and a2x1 + b2y1 + c2 = 0 …(4)
Solving (3) and (4) by the rule of cross multiplication.
x1 y1 1 b 1 c 2−b 2 c 1 and, c 1 a 2−c 2 a1
= = Therefore , x 1= y 1=
b 1 c 2−c 2 b 1 c 1 a 2−c 2 a 1 a1 b 2−a 2 b1 a 1 b 2−a 2 b1 a 1 b 2−a 2 b 1

Again, the point of intersection of (1) and (2) is

( x 1 , y 1 )=¿)

Equation of any Line through Intersection of Two given lines.

Let, a1x + b1y + c1 = 0 …(1)

and, a2x + b2y + c2 = 0 …(2)

Let us consider the equation

(a1x + b1y + c1 = 0) + λ(a2x + b2y + c2) =0 …(3)

where λ is an arbitrary constant. (3) is of first degree in x and y, represents a straight line. The point
of intersection of (1) and (2) which satisfy (1) and (2) simultaneously, also satisfy (3). Therefore, (3)
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represents a straight line passing through the point of intersection of (1) and (2). Here, λ is an
arbitrary constant. For different values of λ, the equation (3) represents different straight lines, all
passing through the intersection of (1) and (2).

What are concurrent lines?

Concurrent lines are three or more straight lines that meet at a single point, known as the point of
concurrency. These lines are always non-parallel, meaning they are not equidistant and eventually
cross. For lines to be considered concurrent, at least three must intersect at the same point. If only
two lines meet at a point, they are simply called intersecting lines, not concurrent. Concurrent lines
are common in geometry and are used to find special points like centroids or incenters in triangles,
where three or more line segments intersect at one location.

The Condition of Concurrency

Let the equations of the three straight lines be,
a1x + b1y + c1 = 0 …(1)
a2x + b2y + c2 = 0 …(2)

and, a3x + b3y + c3 = 0 …(3)

The co-ordinate of the point of intersection of the lines (1) and (2) is
¿)

Lines (1), (2) and (3) will be concurrent if the line (3) also passes through the point of intersection of
the lines (1) and (2).

i.e, if a3(
b 1 c 2−b 2 c 1
a 1 b 2−a 2 b 1
¿+b3(a1b2– a2b1 )
c 1 a 2 – c 2a 1
+ c3 = 0
or, a3(b1c2-b2c1) + b3 (c1a2-a1c2)+c3(a1b2-a2b1) = 0
or,
=0

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Angle Between the Lines

Let y = m1x + c1, and y = m2x + c2 be the equations of the two straight lines AB and AC respectively
intersecting at A and meeting x-axis at the points B and C respectively. Let ϴ1, and ϴ2, be the angles
made by AB and AC with the positive x-axis having slopes m1 = tan ϴ1, and m2 = tanϴ2. If Φ an
acute angle between the lines then ∠ BAC = Φ,

Φ + ϴ2 = ϴ1 [ In △ABC, Exterior angle = sum of opposite interior angles]

From the adjoining figure, we have

Or, Φ = ϴ1 - ϴ2
Now,
tanΦ = tan(ϴ1 - ϴ2)
tanϴ1-tanϴ2
=
1+tanϴ1tanϴ2

m 1−m 2
Therefore, tanΦ = …(1)
1+ m1 m2
The other angle between the lines is π-Φ (obtuse angle between the lines).
For this angle

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 −m1−m2
tan(π – Φ) = -tan Φ = , [from (1)] …(2)
1+m 1m 2
Combining (1) and (2) it follows that
m1−m2
tan Φ = ± …(3)
1+m 1 m2
m1−m2
Φ = tan-1(± ) …(4)
1+ m1 m2

which gives the angles between the two lines. The positive value of tan gives the acute angle
between two lines and the negative value gives the obtuse angle between two lines.

Length of Perpendicular

The Length of the Perpendicular Form a Given Straight Line

Let (x1,y1) be a given point and AB the given straight line, whose equation is
xcosα +ysinα =p.
Draw PM perpendicular to AB, PM=d(say) and draw a line CD parallel to AB and passing through P.
Let p ʹ be the length of the perpendicular from origin on CD, then p ʹ =p+d , if P and the origin are on
opposite side of AB, and then p ʹ =p-d , if P and the origin are on same side of AB. Therefore,
equation of CD is
xcosα +ysinα -(p±d)=0

As P(x1,y1)lies on CD, we have x1cosα +y1sinα -(p±d)=0

or, ±d=x1cosα +y1sinα -p
∴ d=±(x1cosα +y1sinα -p) …(1)
d¿0, If P and origin are on opposite sides and d¿0, if P and origin are on same side.

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Remarks

The length of the perpendicular from a given point(x1,y1) to the given line ax+by+c=0, can be
obtained by reducing it in the form xcosα +ysinα =p.
Here, ax+by+c=0
a b c
x+ y+ [∵Dividing both sides by√ a2 +b 2]
√ a +b
2 2
√ a +b
2 2
√ a +b
2 2

is in the form xcosα +ysinα =p, where

a b c
cosα = , sinα = , -p=
√ a +b
2 2
√ a +b 2 2
√ a +b2
2

Therefore, the length of the perpendicular d from the point (x1,y1) to the line ax+by+c=0 is
±a ±b ±c
x1+ y1+
√ a +b
2 2
√ a +b
2 2
√ a2 +b2
ax 1+by 1+c
=± [Substituting cosα, sinα and p in (1)]
√ a 2 + b2

Also , the distance between the parallel lines ax+by+c1=0 and ax+by+c2=0 is given by
c 2−c 1
d=| |
√ a 2 + b2

The Two Sides of a Line

Let P(x1, y1) and Q(x2, y2) be any two points and let AB be a straight line whose equation is
ax + by + c = 0 …(1)Y
B Y B
P(x1,y1) P(x2,y2)

Q(x2,y2)
R
R
Q(x2,y2)

X
O A O
Fig 3. A Fig 4.
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Join PQ. Let PQ (produced if necessary) meet the line AB at R. Let PR : RQ =m: n. Then by section
formula, the coordinates of R is given by,
¿)

The point lies on the line AB. So it must satisfy the equation (1). Then we obtain

a ( mxm+2+ nxn 1 )+ b( mym+n

2+ny 1
)+c=0
Or, m(ax2+by2+c) + n(ax1+by1+c) = 0
ax 1+ by 1+c −m
= …(2)
ax 2+ by 2+c n
Case I : Let P∧Q are on opposite sides of the line AB . Then R∣PQ internally , Fig 1 ,∈the ratio m:n∧therefore , m
ax 1+ by 1+c
Thisimplies that ax 1+by1+c and ax2+by2+c are opposite in sign.
ax 2+ by 2+c

Case II: Suppose P and Q are on the same side of line AB, Fig 2. Then R divides PQ
externally, in
< 0the ratio m:n therefore m/n is negative. So, from (2) it follows that
ax 1+ by 1+c
This implies that ax1+by1+c and ax2+by2+c are of same sign.
ax 2+ by 2+c

>0

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 Equations of the bisectors

Fig 5

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Pair of Straight Lines

Homogeneous Equations

An equation in two variables x and y is said to be homogeneous if the sum of the powers of x and y
in each term is always the same. If the sum is two, then the equation is called a homogeneous
equation of degree two in x and y or a second degree homogeneous equation.

Examples:

The above equations (1),(2),(3),and (4) represent homogeneous equations of degree n, degree 3,
degree 2, and degree 2, respectively. The equation of degree n is also called an nth-degree equation.
Thus, (1),(2),(3),and (4)(1), (2), (3), and (4)(1),(2),(3),and (4) are called nth, 3rd 2nd and 2nd degree
homogenous equations

Equations of Bisectors represented by ax2+2hxy+by2=0

If the equation
ax2+2hxy+by2=0 …(1)
represents the lines
y - m2x = 0 …(2)
and, y – m2x = 0, …(3)
so that,
m1 + m2 = -2h/b and m1m2 = a/b.

The equations of bisectors of (2) and (3) are

y−m1 x y−m2 x
=±
√1+ m 1
2
√1+m 2
2

Taking positive sign,

y−m1 x + y−m2 x
= …(4)
√1+ m 1
2
√ 1+ m 2
2

Taking negative sign,

y−m1 x − y−m2 x
= …(5)
√1+ m 1
2
√ 1+ m 2
2

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Solving (4) and (5) we get,

2 2
x − y xy
= …(6)
a−b h
as equations of bisectors.

General Equation of Second Degree

Let a1x+b1y+c1 = 0 … (1)
and a2x+b2y+C2 = 0 … (2)
be equations of any two straight lines. Consider their combined equation
(a1x+b1y+c1) (a2x+b2y+c2)=0 ...(3)
The coordinates of any point on the locus of equations (1) or (2) satisfy (3) Similarly on pages co-
ordinates of any point on the locus of (3) must satisfy either equation (1) or (2) or both. Thus, (3)
represents the pair of lines represented by equation (1) and (2). On simplification, equation (3) can be
written as
a1a2x2+(a1b2+a2b1)xy+b1b2y2+(a1c2+a2c1)x+(b1c2+b2c1)y+c1c2=0,
or, ax2+2hxy + by2+2gx+2fy + c = 0, _(4)
is a general equation of second degree in x and y, where
a = a1a2, b = b1b2, C = C1C2,
2h = a1b2+a2b1, 2g= a1c1+ a2c1, 2f=b1c2+ b2c1

So, the pair of lines is always represented by a general equation of second degree. But the converse is
not true. That is, a general equation of second degree does not necessarily always represent a pair of
straight lines.

x2+y2−r2=0, is a general equation of second degree. But it represents a circle with center at the origin
and radius r.

Condition for the general equation of second degree to represent a pair of lines:

The general equation of second degree in x and y is

ax2+2hxy+by2+2gx+2fy+c=0, a≠0
or,ax2+2x(hy+g)+by2+2fy+c=0 _(1)
Solving for x , we get
1
x= {−(hy+g)±√(hy+ g)2−a(by 2 +2 fy+ c)},a≠0
a
Equation (1) will represent two lines if
(hy+g)2 −a(by2+2fy+c) is a perfect square.
(h2+ab) y2−2(hg-fa) y+g2-ac is a perfect square.

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To be a perfect square, its discriminant must be zero.
i.e., (gh - af)2 +(h2 - ab) (g2 - ac) = 0 ).
or, a (abc + 2fgh – af2 – bg2 – ch2 = 0 ).
Since a ≠ 0 , the required condition is
abc + 2fgh – af2 – bg2 – ch2 = 0
The condition can also be written in the form

| |
a h g
h b f =0
g f c

2. If the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of lines, then ax2 + 2hxy +
by2 = 0 represents a pair of lines through the origin parallel to the above pair.
Let the lines represented by ( ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ) be
L1x + m1 y + n1 = 0 and l2 x + m2 y + n2 = 0
Then, we have
Ax2 + 2hxy + by2 + 2gx + 2fy + c = (l1 x + m1 y + n2)(l2 x + m2 y + n2) = 0 _(1)
Comparing the coefficients of similar terms in both sides of (1), we get
L1 l2 = a, m1 m2 = b, n1 n2 = c,
M1n2 + m2 n1 = 2f, n1l2 + n2l1 = 2g, _ (2)
L1 m2 + l2 m1 = 2h
Now, the lines through the origin parallel to the above lines are (l1 x + m1 y) = 0) and (l2 x + m2 y) = 0
and their combined equation is

(l1x + m1y) (l2x + m2y) = 0

L1l2 x2 + xy (l1m2 + l2m1) + m1m2 y2 = 0
i.e., ax^2 + 2hxy + by^2 = 0
which represents a pair of lines through the origin parallel to the above pair.
Also, angles between the line pair
Ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 _ (3)
are same as the angles between the line pair
ax2 + 2hxy + by2= 0 _ (4)
as the lines represented by (3) and (4) are parallel.
So, the angles are given by
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 2 √ h2 – ab
tan θ = ±
a+b

2 √ h2 – ab
i.e., θ = tan−1± ( )
a+b
Hence, the two lines represented by (3) will be perpendicular to each other if a + b = 0 and they will
be parallel and coincidence if h2 = ab.
3. Equation to the pair of lines joining the origin to the point of intersection of the line (lx + my = n)
and the curve ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.
The given general equation of second degree is
Ax2+2hxy+by2+2gx+2fy+c=0 _ (1)
Equation (1) generally represents a curve except in case of pair of straight lines. Let the straight line
lx+my=n _ (2)

meet the curve, (Fig above), in two points P and Q. We have to find the equation of OP and OQ.
Equation (2) may be written as:
lx+my
=1 _ (3)
n
Making the equation (1) the homogeneous equation of second degree in and, with the help of (3), we
get

lx+my
ax2 + 2hxy + by2 + 2(gx+fy)( ¿+ c ¿)2 =0 _(4)
n
Equation (4) being second degree homogeneous in and , represents a pair of straight lines through
the origin. Also, the coordinates of the point of intersection of the straight line (3) and the curve (1)
satisfy both the equations (1) and (2) and hence, equation (4). Therefore, equation (4) is the equation
of pair of lines joining the origin to the points of intersection of line (2) and curve (1).
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Conclusion
Through this project work, a wide range of concepts related to straight lines was explored and
understood in greater depth. The process began with learning how to reduce a general straight-line
equation into three important standard forms, including the slope-intercept form, the intercept form,
and the normal (or perpendicular) form. Understanding these forms provided a strong foundation for
analyzing and interpreting different types of lines and their behaviors in the coordinate plane.

In addition, the project covered how to calculate the length of a perpendicular drawn from a point to
a line, which is an essential concept in geometry and analytical mathematics. This involved using
formulas and applying geometric reasoning to determine shortest distances.

Another important aspect studied was how straight lines intersect or bisect each other. By working
with such interactions, we learned how to find the equations of lines that divide angles or segments,
as well as how to determine the point of intersection between two lines.

Overall, the project enhanced our understanding of linear equations, their geometric representations,
and their practical applications. It also improved problem-solving skills and the ability to apply
mathematical concepts to real-life situations involving lines and distances.

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