In Fig. 9.18, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS
Solution:
Given, PQ and PR are the tangents drawn to a circle.
Also, ∠RPQ = 30°
A chord RS is drawn parallel to the tangent PQ.
We have to find the measure of angle RQS.
We know that the tangents drawn through an external point to a circle are equal.
So, PQ = PR.
We know that the angles opposite to the equal sides are equal in a triangle.
So, ∠PRQ = ∠PQR --------------- (1)
Considering triangle PQR,
We know that the sum of all three interior angles of a triangle are equal.
∠PQR + ∠RPQ + ∠PRQ = 180°
∠PQR + 30° + ∠PRQ = 180°
∠PQR + ∠PRQ = 180° - 30°
∠PQR + ∠PRQ = 150°
From (1),
∠PQR + ∠PQR = 150°
2∠PQR = 150°
∠PQR = 150°/2
∠PQR = 75°
Since PQ = PR, ∠PQR = ∠PRQ = 75°
We know that the angle between the tangent and the chord of a circle is equal to the angle made by the chord in the alternate segment.
So, ∠PQR = ∠RSQ = 75° ------------------- (2)
Since RS || PQ,
The alternate angles are equal.
i.e., ∠PQR = ∠QRS = 75° ------------------ (3)
Comparing (2) and (3),
∠RSQ = ∠QRS = 75°
We know that the sides opposite to equal angles are equal.
So, SQ = RQ
This implies QRS is an isosceles triangle.
In triangle QRS,
∠RSQ + ∠QRS + ∠RQS = 180°
75° + 75° + ∠RQS = 180°
150° + ∠RQS = 180°
∠RQS = 180° - 150°
Therefore, ∠RQS = 30°
✦ Try This: If PT is a tangent to a circle with center O and PQ is a chord of the circle such that ∠QPT = 70°, then find the measure of ∠POQ.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 7
In Fig. 9.18, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS
Summary:
In Fig. 9.18, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. The measure of ∠RQS is 30°
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