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

∆QRS is an equilateral triangle.   ...(Given)

∴ seg RS ≅ seg QS ≅ seg QR   ...(Sides of an equilateral triangle)

Arcs of the same circle are equal, if the related chords are congruent.

∴ arc RS ≅ arc QS ≅ arc QR.

Let m(arc RS) = m(arc QS) = m(arc QR) = x   ...(1)

m(arc RC) + m(arc QS) + m(arc QR) = 360°   ...(Measure of a circle is 360°)

∴ x + x + x = 360°

∴ 3x = 360°

∴ x = `360/3`

∴ x = 120°

∴ m(arc RS) = m(arc QS) = m(arc QR) = 120°

m(arc QRS) = m(arc QR) + m(arc RS)   ...(Arcs addition postulate)

∴ m(arc QRS) = 120° + 120°

∴ m(arc QRS) = 240°