Question

In the adjoining figure, AQRS is an equilateral triangle. Prove that, 

i. arc RS ≅ arc QS ≅ arc QR 

ii. m(arc QRS) = 240°.

Answer 1

Proof: 

i. ∆QRS is an equilateral triangle, [Given] 

∴ seg RS ≅ seg QS ≅ seg QR [Sides of an equilateral triangle] 

∴ arc RS ≅ arc QS ≅ arc QR [Corresponding arcs of congruents chords of a circle are congruent] 

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

m(arc RS) + m(arc QS) + m(arc QR) = 360° [Measure of a circle is 360° , arc addition property] 

∴ x + x + x = 360° 

∴ 3x = 360° 

∴ x = 360°/3 = 120° 

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

Now, m(arc QRS) = m(arc QR) + m(arc RS) [Arc addition property]

= 120° + 120° [From (i)] 

∴ m(arc QRS) = 240°