Question

3) PQRS is a cyclic quadrilateral,
Side PQ = side QR, ZPSR =110°
Find : i) ZPOR
sk 110°
ii) m (arc PQR)
iii) m (arc QR)
R​

Answer 1

Answer:

Given:-

• PQRS is a cyclic quadrilateral
• PQ = QR
• ∠PSR = 110°

To Find:-

• ∠PQR
• m (arc PQR)
• m (arc QR)

Solution:-

(i) It is given that PQRS is a cyclic quadrilateral

• opposite angles of a cyclic quadrilateral are supplementary

⇒ ∠PSR + ∠PQR = 180°

⇒ 110 + ∠PQR = 180°

⇒ ∠PQR = 180 - 110

⇒ ∠PQR = 70°

(ii) 2 × ∠PQR = m (arc PR) [the measure of an inscribed angle is half the measure of the arc intercepted by it].

m (arc PR) = 140°

m (arc PQR) = 360° - 140° = 220° [using measure of a major arc = 360° - measure of it's corresponding minor arc].

(iii) Side PQ = Side RQ

m (arc PQ) = m (arc RQ) [corresponding arcs of congruent chords of a circle are congruent]

⇒ m (arc PQR) = m (arc PQ) + m (arc RQ)

⇒ m (arc PQR) = 2 × m (arc PQ)

⇒ m (arc PQ) = 110°

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