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Grade IX

Mathematics

Geometry Notes

Geometry Notes

● The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the

distance of the point from the x-axis is called its y-coordinate, or ordinate.

● If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of

the point.

● The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the

y-axis are (0, y). ● The coordinates of the origin are (0, 0)

● The coordinates of a point are of the form (+ , +) in the first quadrant, (–, +) in the second

quadrant, (–, –) in the third quadrant and (+, –) in the fourth quadrant, where + denotes a

positive real number and – denotes a negative real number.

● Some of Euclid’s axioms were :

(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.

● Euclid’s postulates were :

Postulate 1 : A straight line may be drawn from any one point to any other point.

Postulate 2 : A terminated line can be produced indefinitely.

Postulate 3 : A circle can be drawn with any centre and any radius.

Postulate 4 : All right angles are equal to one another.

Postulate 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 angles is less than two right

angles

Geometry Notes

● Angle sum property of triangles: The sum of all the three interior angles of a triangle is 180°. ∴ ∠A + ∠B + ∠C = 180°

● Facts deduced from angle sum property of triangles:

- There can be no triangle with two right angles or two obtuse angles. - There can be no triangle with all angles less than or greater than 60°.

● Relation between the vertex angle and the angles made by the bisectors of the

remaining angles:

In ΔABC, BX and CY are bisectors of ∠B and ∠C respectively. Also, O is the point of intersection of BX and CY.

Therefore, .

● If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of

the two interior opposite angles.

Geometry Notes

∠ACX = ∠BAC + ∠ABC This property is known as exterior angle property of a triangle.

● Corresponding angles axiom If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

∠1 =∠5, ∠2 =∠6, ∠3 =∠7 and ∠4 = ∠8

● Converse of corresponding angles axiom If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Geometry Notes

In the figure, the corresponding angles are equal. Therefore, the lines l and m are parallel to each other.

● Alternate angles axiom If a transversal intersects two parallel lines, then the angles in each pair of alternate angles

are equal.

In the above figure, lines l and m are parallel. So, by using the alternate angles axiom, we

can say that:

∠1 = ∠7, ∠2 = ∠8, ∠3 = ∠5 and ∠4 = ∠6

● Converse of alternate angles axiom If a transversal intersects two lines such that the angles in a pair of alternate angles are equal, then the two lines are parallel.

In the above figure, alternate interior angles are equal (100°) and thus, lines l and m are parallel.

Geometry Notes

● Property of interior angles on the same side of a transversal:

If a transversal intersects two parallel lines, then the angles in a pair of interior angles on the

same side of the transversal are supplementary.

In the given figure, if lines l and m are parallel to each other then ∠1 + ∠4 = 180° and ∠2 + ∠3 = 180°.

● Converse of the property of interior angles on the same side of a transversal:

If a transversal intersects two lines such that the interior angles on the same side of the transversal are supplementary, then the lines intersected by the transversal are parallel.

● Lines which are parallel to the same line are parallel to each other. In the given figure, AB || CD and CD || EF, therefore AB || EF.

●

Pair of Angles Condition

Complementary Angles Measures add up to 90°

Supplementary Angles Measures add up to 180°

Adjacent Angles Have common vertex and common arm

Geometry Notes

but no interior common.

Linear Pair Adjacent and supplementary

● SAS congruence rule

If two sides of a triangle and the angle included between them are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent by SAS congruence rule.

● ASA congruence rule

If two angles and included side of a triangle are equal to the two corresponding angles and the included side of another triangle, then the triangles are congruent by ASA congruence rule.

● RHS congruence rule

If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side

of the other right triangle, then the two triangles are congruent to each other by RHS

congruence rule.

● SSS congruence rule

If three sides of a triangle are equal to the three sides of the other triangle, then the two

triangles are congruent by SSS congruence rule.

● Sum of the angles of a quadrilateral is 360°.

● Opposite sides in a parallelogram are equal. Conversely, in a quadrilateral, if each pair of

opposite sides are equal then the quadrilateral is a parallelogram.

● Diagonal of a parallelogram divides it into two congruent triangles. In the given figure, if ABCD is a parallelogram and AC is its diagonal then ΔABC ΔCDA.

● In a parallelogram, opposite angles are equal. Conversely in a quadrilateral, if pair of

opposite angles is equal, then the quadrilateral is a parallelogram.

Geometry Notes

● The diagonals of a parallelogram bisect each other. Conversely, if the diagonals of a

quadrilateral bisect each other, then it is a parallelogram. Suppose ABCD is a quadrilateral. The diagonals of the quadrilateral intersect at O such that AO = OC and DO = OB

Therefore, ABCD is a parallelogram.

● A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. ● Mid-point theorem states that the line segment joining the mid-point of any two sides of a

triangle is parallel to the third side and is half of it.

In ΔABC, if D and E are the mid - points of sides AB and AC respectively then by mid -

point theorem DE ॥ BC and DE = ● Converse of the mid-point theorem is also true, which states that a line through the mid -

point of one side of a triangle and parallel to the other side bisects the third side.

In ΔABC, if AP = PB and PQ ॥ BC then PQ bisects AC i.e., Q is the mid - point of AC.

Geometry Notes

Properties of Geometrical Shapes

NAME SHAPE PROPERTIES

Square

1. It has four sides and four vertices. 2. All the sides are of equal length. 3. The measure of each angle is 90°. 4. It has 2 diagonals. 5. Diagonals are of equal length.

Rectangle

1. It has four sides and four vertices. 2. Opposite sides are equal in length. 3. The measure of each angle is 90°. 4. It has two diagonals. 5. Diagonals are of equal length.

Parallelogram

1. Opposite sides of are of equal length. 2. Opposite sides are parallel. 3. Opposite angles are of equal measure. 4. Adjacent angles are supplementary. 5. Diagonals bisect each other. 6. Sum of all the interior angles is 360°.

Geometry Notes

Rhombus

1. All sides are of equal length. 2. Opposite sides are parallel. 3. Opposite angles are equal. 4. Diagonals are perpendicular bisectors of each other. 5. Sum of all the interior angles is 360°.

Trapezium

1. One pair of sides is parallel. 2. Adjacent angles made by non parallel side are supplementary. 3. Sum of all the interior angles is 360°.

Kite

1. Two distinct pairs of adjacent sides are of equal length. 2. Diagonals are perpendicular to each other. 3. Longer diagonal bisects the shorter one. 4. Sum of all the interior angles is 360°.

● Parallelograms on the same (or equal) base and between the same parallels are equal in area.

Converse of the property is also true, which states that parallelograms on the same base and having

equal areas lie between the same parallels.

● If a parallelogram and a triangle lie on the same (or equal) base and between the same

parallels then the area of the triangle is half the area of the parallelogram.

Geometry Notes

● Triangles on the same base (or equal base) and between the same parallels are equal in area.

In the given figure, ΔABC and ΔDBC lie on the same base BC and BC || XY, therefore area of ΔABC is equal to the area of ΔDBC.

Converse of the property is also true, which states that triangles having the same base and

equal areas lie between the same parallels.

● An important result states that a median of a triangle divides it into two triangles of equal

area.

● Chords of a circle that are equal in length subtend equal angles at the centre of the circle. In the given figure, if AB and CD are two equal chords then ∠AOB = ∠COD

Converse of this property also holds true, which states that chords subtending equal angles at the centre of the circle are equal in length.

● Perpendicular drawn from the centre of a circle to a chord bisects the chord. In the given figure, AL will be equal to LB if OL ⊥ AB, where O is the centre of the circle.

Geometry Notes

Converse of this property also holds true, which states that the line joining the centre of the circle to the mid-point of a chord is perpendicular to the chord.

● There is one and only one circle passing through three given non-collinear points.

Therefore, at least three points are required to construct a unique circle. ● Equal chords of a circle (or congruent circles) are equidistant from the centre of the circle. In the given figure, OL will be equal to OM if AB = CD, where O is the centre of the circle.

Converse of the property also holds true, which states that chords which are equidistant from the centre of a circle are equal in length.

● If two chords of a circle are equal then their corresponding arcs (minor or major) are

congruent. In the given figure, arc AB will be congruent to arc CD if chord AB = chord

CD.

Converse of the property also holds true, which states that if two arcs of a circle are congruent then their corresponding chords are equal.

● Congruent arcs subtend equal angles at the centre of the circle. In the given figure, ∠AOB

will be equal to ∠COD if arcs AB and CD are congruent.

Geometry Notes

Converse of the property is also true, which states that two arcs subtending equal angles at the centre of the circle are congruent.

● The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at the remaining part of the circle.

● The angle lying in the major segment is an acute angle and the angle lying in the minor segment is an obtuse angle. This statement is true for all major and minor segments in a circle.

● Angles in the same segment of a circle are equal. In the given figure, ∠PRQ and ∠PSQ lie in the same segment of the circle. ∴ ∠PRQ = ∠PSQ

● A set of points that lie on a common circle are known as concyclic points.

Here, points A, B, D and E are concyclic points.

● If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment then the four points are concyclic.

● The sum of each pair of opposite angles of a cyclic quadrilateral is 180°.

Geometry Notes

● Converse of the property also holds true, which states that if the sum of a pair of opposite angles of a quadrilateral is 180° then the quadrilateral is cyclic.

● Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

● In case of a triangle ABC, with sides of lengths a, b and c units:

Perimeter of ABC = AB + BC + AC = a + b + c

● The semi-perimeter of a triangle is half the perimeter of the triangle.

The semi-perimeter (s) of a triangle with sides a, b and c is .

● Area of triangle using Heron's formula:

When all the three sides of a triangle are given, its area can be calculated using Heron’s formula, which is given by:

Area of triangle = ● Surface areas of cuboid:

Lateral surface area of the cuboid = 2h (l + b) Total surface area of the cuboid = 2 (lb + bh + hl)

Note: Length of the diagonal of a cuboid =

Geometry Notes

● Surface areas of cube:

Lateral surface area of the cube = 4a2

Total surface area of the cube = 6a2

Note: Length of the diagonal of a cube =

● Surface areas of solid cylinder

○ Curved surface area = 2πrh, where r and h are the radius and height

○ Total surface area = 2πr (r + h), where r and h are the radius and height

● Volume of cube and cuboid

○ Volume of cube = a3, where a is the side of the cube

○ Volume of cuboid = l × b × h, where l, b and h are respectively the length, breadth

and height of the cuboid.

● Volume of the solid cylinder and hollow cylinder

○ Volume of solid cylinder = πr2h, where r and h are the radius and height of the solid

cylinder

Geometry Notes

○ Volume of the hollow cylinder = π (R2 − r2) h, where r, R and h are the inner radius,

outer radius and height of hollow cylinder

● Relationship between common units of volume and capacity:

1 ml = 1 cm3

1 l = 1000 cm3

1 m3 = 1000000 cm3 = 1000 l

● Volume of sphere and hemisphere

○ Volume of sphere

○ Volume of hemisphere

● Volume of a cone = , where r and h are the radius of base and height of the cone.

● Surface areas of sphere and hemisphere

○ Surface area of sphere = 4πr2, where r is the radius

Geometry Notes

○ Curved surface area of hemisphere = 2πr2, where r is the radius

○ Total surface area of hemisphere = 3πr2, where r is the radius

○ curved surface area of hollow hemisphere=

○ Total surface area of hollow hemisphere =

● Surface areas of cone

○ Curved surface area = πrl, where r and l are the radius and slant height

○ Total surface area = πr (l + r), where r and l are the radius and slant height

Here, , where h is the height.