Question

In the given figure, ∆QRS is an equilateral triangle. Prove that,
(1) arc RS ≅ arc QS ≅ arc QR
(2) m(arc QRS) = 240°.

Answer 1

(1)
It is given that ∆QRS is an equilateral triangle.

∴ chord RS = chord QS = chord QR (Sides of an equilateral triangle are equal)

⇒ m(arc RS) = m(arc QS) = m(arc QR) .....(1) (Corresponding arcs of congruent chords of a circle are congruent)

(2)
m(arc RS) + m(arc QS) + m(arc QR) = 360º (Measure of a complete circle is 360º)

⇒ m(arc RS) + m(arc RS) + m(arc RS) = 360º [Using (1)]

⇒ 3 × m(arc RS) = 360º

⇒ m(arc RS) = 120º

∴ m(arc RS) = m(arc QR) = 120º

Now,

m(arc QRS) = m(arc QR) + m(arc RS)

⇒ m(arc QRS) = 120º + 120º = 240º