Formulas

Laws of Exponents:
(i) am × an
"#
(ii) = am-n
"$
(iii) (𝑎m)n = amn
(iv) am × bm = (ab)m
(v) (ab)m = ambm
" "#
(vi) (' )m =
'#
(
(vii) a-n = "$
(viii) a0 = 1

Expansions:
(i) (a+b)2 = (a2+2ab+b2)
(ii) (a-b)2 = (a2-2ab+b2)
(iii) (x+a)(x+b) = x2 + x(a+b) + ab
(iv) (a+b+c)2 = a2 + b2 + c2 + 2(ab+bc+ca)
(v) (a+b)3 = a3 + b3 + 3ab(a+b)
(vi) (a-b)3 = a3 - b3 - 3ab(a-b)
(vii) a2 – b2 = (a+b)(a-b)
(viii) a3 + b3 = (a+b)(a2 – ab + b2)
(ix) a3 – b3 = (a-b)(a2 + ab + b2)
(x) x3 + y3 + z3 -3xyz = (x+y+z)(x2 + y2 + z2 – xy – yz – zx)
(xi) (x+y+z) = 0 → (x3 + y3 + z3) = 3xyz
(xii) √𝑎𝑏 = √𝑎√𝑏
" √"
(xiii) √' =
√'
(xiv) (√𝑎 + √𝑏) (√𝑎 − √𝑏) = a – b
(xv) (𝑎 + √𝑏)(𝑎 − √𝑏) = a2 – b
(xvi) 0√𝑎 + √𝑏10√𝑐 + √𝑑1 = √𝑎𝑐 + √𝑎𝑑 + √𝑏𝑐 + √𝑏𝑑)
(xvii) 0√𝑎 + √𝑏12 = 𝑎 + 2√𝑎𝑏 + 𝑏

"7'78
Heron’s Formula = 5𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where s =
9

Solid Curved/Lateral surface Total Area Volume

area
Cuboid 2h(l+b) 2(lb+bh+hl) l.b.h
Cube 4a2 6a2 a3
Right Circular Cylinder 2𝜋rh 2𝜋r(r+h) 𝜋r2h
(closed at top)
Right Circular Cylinder 2𝜋rh 𝜋r(2h+r) 𝜋r2h
(open at top)
Hollow Cylinder 2𝜋rh(R + r) 2𝜋h(R + r)(h + R – r) 𝜋h(R2 – r2)
Cone 𝜋rl 𝜋r(r+l) (
;
𝜋r2h
Sphere 4𝜋r2 4𝜋r2 <
;
𝜋r3
Hemisphere 2𝜋r2 3𝜋r2 9
;
𝜋r3

Diagonal of cuboid = √𝑙 9 + 𝑏9 + ℎ 9 Diagonal of cube = √3a

Slant Height, l = √ℎ 9 + 𝑟 9

ABBCD EFGFH7EIJCD EFGFH KFLFGMG 8N"OO OFPC × QDCRMCL8S

Class Mark = 9
Adjusted Frequency = TN"OO OFPC
Euclid’s Geometry
Definitions:
1. A point is that which has no part.
2. A line is breathless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.

Axioms (common notions; assumptions; not specific to geometry):

1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.
5.1 Given two distinct points, there is a unique line that passes through them.

Postulates (assumptions specific to geometry):

1. Given two distinct points, there is a unique line that passes through them.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken
together less than two right angles, then the two straight lines, if produced indefinitely, meet on
that side on which the sum of the angles is less than two right angles.

Theorem 5.1 Two distinct lines cannot have more than one point in common.

Lines & Angles

1. Complementary Angles: A pair of angles whose sum of 90°.
2. Supplementary Angles: A pair of angles whose sum is 180°.
3. Adjacent Angles: Pair of angles having a common vertex and a common arm.
4. Linear Pair: Adjacent angles whose sum is 180°.
5. Vertically Opposite Angles: Opposite angles formed when two lines intersect each other.

Axioms:
6.1 If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.
6.2 If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
6.3 Lines which are parallel to the same lines are parallel to each other.

Theorem 6.1 If two lines intersect each other, then the vertically opposite angles are equal.

Triangles
Axioms:
7.1 (SAS Congruence Rule): Two triangles are congruent if two sides and the included angle of one angle
are equal to the two sides and the included angle of the other triangle.

Theorems:
7.1 (ASA Congruence Rule): Two triangles are congruent if two angles and the included side of one angle
are equal to the two angles and the included side of the other triangle.
7.2 Angles opposite to equal sides of an isosceles triangle are equal.
7.3 The sides opposite to equal angles of a triangle are equal.
 7.4 (SSS Congruence Rule): If three sides of one triangle are equal to the three sides of another triangle,
then the two triangles are congruent.
7.5 (RHS Congruence Rule): If in two right triangles, the hypotenuse and one side of one triangle are equal
to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

Quadrilaterals
Properties of Quadrilaterals:
Parallelogram:
1. Opposite sides are equal and parallel.
2. Opposite angles are equal.
3. Diagonals bisect each other.
4. Adjacent angles are supplementary.

Rectangles:
1. Opposite sides are equal and parallel.
2. Opposite angles are equal.
3. Diagonals are equal and bisect each other.
4. Adjacent angles are supplementary.
5. All angles are right angles.
Kite:
1. Two disjoint pairs of consecutive sides are congruent.
2. Diagonals bisect each other at right angles.
3. One of the diagonals bisects a pair of opposite angles.
4. One pair of opposite angles are congruent.

Rhombus:
1. Opposite sides are parallel.
2. All sides are equal.
3. Opposite angles are equal.
4. Diagonals bisect each other at right angles.
5. The diagonals bisect the angles.
6. The diagonals divide the rhombus into four congruent
triangles.
Squares:
1. Opposite sides are parallel.
2. All sides are equal.
3. All angles are equal and 90° each.
4. Diagonals are equal and bisect each other at right angles.
5. The diagonals bisect the angles.
6. The diagonals form four congruent isosceles triangles.

Trapeziums:
1. One pair of opposite sides are parallel.
2. Adjacent angles on non-parallel sides are supplementary.

Isosceles Trapezium:
1. Legs are congruent.
2. Bases are parallel.
3. Lower base angles are congruent.
4. Upper base angles are congruent.
5. Diagonals are congruent.
6. Any lower base angle is supplementary to any upper base
angle.
Theorems:
8.1 A diagonal of a parallelogram divides it into two congruent triangles.
8.2 In a parallelogram, opposite sides are equal.
8.3 If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
8.4 In a parallelogram, opposite angles are equal.
8.5 If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
8.6 The diagonals of a parallelogram bisect each other.
8.7 If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
8.8 (Mid Point theorem): The line segment joining the mid-points of two sides of a triangle is parallel to the
third side and half of it.
8.9 (Converse of Mid Point theorem): The line drawn through the mid-point of one side of a triangle,
parallel to another side, bisects the third side.

Circles
Theorems
9.1 Equal chords of a circle subtend equal angles at the centre.
9.2 If the angles subtended by the chords of a circle at the
centre are equal then, the chords are equal.
9.3 The perpendicular from the centre bisects the chord.
9.4 The line drawn through the centre of a circle to bisect a
chord is perpendicular to the chord.
9.5 Equal chords of a circle are equidistant from the centre.
9.6 Chords equidistant from the centre are equal in length.
9.7 The angle subtended by an arc at the centre is double the angle subtended by it at any point on the
remaining part of the circle.
9.8 Angles in the same segment are equal.
9.9 If the line segment joining two points subtends equal angles at two other points lying on the same side
of the line containing the line segment, the four points lie on a circle (they are concyclic).
9.10 The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
9.11 If the sum of a. pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.