Q2 of 59 Page 63

In fig 3.38 ∆ QRS is an equilateral triangle. Prove that,

(1) arc RS ≅ arc QS ≅ arc QR


(2) m(arc QRS) = 240°.



(1) Two arcs are congruent if their measures and radii are equal.


∵∆ QRS is an equilateral triangle


∴ RS = QS = QR


⇒arc RS ≅ arc QS ≅ arc QR


(2) Let O be the centre of the circle.


m(arc QS) = ∠ QOS


∠ QOS + ∠ QOR + ∠ SOR = 360°


⇒ 3∠ QOS = 360° {∵ ∆QRS is an equilateral triangle}


⇒∠ QOS = 120°


m(arc QS) = 120°


m(arc QRS ) = 360° - 120° {∵Measure of a major arc = 360°- measure of its corresponding minor arc}


⇒m(arc QRS ) = 240°


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