Question

A chord of a circle is equal to its radius. prove that the angle subtended by the chord at a point on the the circumference of the circle is either 30 degree or 150 degree​

Answer 1

Answer:

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle and the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Draw a circle with any radius and center O. Let AO and BO be the 2 radii of the circle and let AB be the chord equal to the length of the radius. Join them to form a triangle.

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.

Here OA = OB = AB

Hence ∆ABO becomes an equilateral triangle.

Draw 2 points C and D on the circle such that they lie on the major arc and minor arc, respectively.

Since ∆ABO is an equilateral triangle, we get ∠AOB = 60°.

For the arc AB, ∠AOB = 2∠ACB as we know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

∠ACB = 1/2 ∠AOB = 1/2 × 60 = 30°

As you can notice the points A, B, C, and D lie on the circle. Hence ABCD is a cyclic quadrilateral.

We know that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Therefore,

∠ACB + ∠ADB = 180°

30° + ∠ADB = 180°

∠ADB = 150°

So, when the chord of a circle is equal to the radius of the circle, the angle subtended by the chord at a point on the minor arc is 150° and also at a point on the major arc is 30°.

☛ Check: NCERT Solutions Class 9 Maths Chapter 10

Video Solution:

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

Maths NCERT Solutions Class 9 Chapter 10 Exercise 10.5 Question 2

Summary:

If a chord of a circle is equal to the radius of the circle, then the angle subtended by the chord at a point on the minor arc is 150° and also at a point on the major arc is 30°.

☛ Related Questions:

In Fig. 10.37, ∠PQR = 100° where P, Q and R are points on a circle with center O. Find ∠OPR.

In Fig. 10.38, ∠ABC = 69° and ∠ACB= 31°, find ∠BDC.

In Fig. 10.39, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC.

ABCD is a cyclic quadrilater