Class 9 Maths

Circle

Exercise 10.5 Part 1

Question 1: In the given figure, A, B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

Circle Segment Sector

Answer: ∠AOC = 60° + 30° = 90°
Since angle made by an arc on the centre is double the angle made anywhere else on the circle
Hence, ∠ADC = 90°/2 = 45°

Question 2: A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Answer: Given a circle with centre O in which chord AB = radius OB

Circle Segment Sector

As OA = OB = AB
So, ΔOAB is equilateral triangle
So, ∠AOB = 60°
Angle made on centre by an arc is double the angle made anywhere else on the circle
Hence, ∠ADB = 30°
As ACBD is a cyclic quadrilateral
Hence, ∠ACB = 180°- ∠ADB
Or, ∠ACB = 180°- 30° = 150°

Question 3: In the given figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

Circle Segment Sector

Answer: Here, PQRS is a cyclic quadrilateral

Circle Segment Sector

Hence, ∠PSR = 180° - ∠PQR
Or, ∠PSR = 180° - 100° = 80°
As angle made by an arc on centre is double than angle made anywhere else
So, ∠POR = 2 x 80° = 160°
In ΔPOR we have PO = RO
So, ∠OPR = ∠ORP
Or, 160° + 2∠OPR = 180°
Or, 2∠OPR = 180° - 160°= 20°
Or, ∠OPR = 10°

4. In the given figure, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.

Circle Segment Sector

Answer:In ΔABC;
∠BAC = 180° - (69° + 31°)
Or, ∠BAC = 180° - 100° = 80°
Angles made on one side of a chord are equal
Hence, ∠BDC = ∠BAC = 80°


Number System

Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers.

Polynomials

Algebraic expression with many terms having variables and coefficients is called polynomial.

Coordinate Geometry

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.

Linear Equations in Two Variables

A linear equation in two variables has infinite number of solutions.

Euclid's Geometry

The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.

Lines & Angles

An angle is formed by two rays originating from the same end point. The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

Triangles

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

Quadrilaterals

Sum of the angles of a quadrilateral is 360°. A diagonal of a parallelogram divides it into two congruent triangles.

Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, and diagonals bisect each other

Circles

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

Heron's Formula

Heron's formula is generally used for calculating area of scalene triangle.

Surface Area

How to calculate surface area of cylinder, cone and sphere?

Volume

How to calculate volume of cylinder, cone and sphere?

Statistics

Learn to make bar chart and histogram from given data.

Probability

Probability tells about likelihood of an event to occur.