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