Question

2) In the given figure,
A QRS is an equilateral
triangle. Prove that :
i) arc RS arc QS = arc QR
ii) m (arc QRS) = 240°.
​

Answer 1

A QRS is an equilateral  triangle. i) arc RS arc QS = arc QR ii) m (arc QRS) = 240°.

• To prove: i) arc RS arc QS = arc QR
•                 ii) m (arc QRS) = 240°.
• Proof:
• i) Given,
• QRS is an equilateral triangle.
• ⇒ RS = QS = QR   (sides of an equilateral triangle are equal)
• ⇒ chord RS = chord QS = chord QR
• ⇒ m (arc RS) = m (arc QS) = m (arc QR)  (corresponding  arcs of corresponding chords are equal)
• ∴arc RS arc QS = arc QR
• ii) We have,
• m (arc RS) + m (arc QS) + m (arc QR) =360° (complete circle = 360°)
• using the result of (i), we have
• ⇒ m (arc RS) + m (arc RS) + m (arc RS) =360°
• ⇒ 3 m (arc RS) =360°
• ⇒ m (arc RS) = 120°
• Now,
• m (arc RS) = 120° = m (arc QR)
• ⇒ m (arc QRS ) = 120° + 120°
• ∴ m (arc QRS ) = 240°

Answer 2

A QRS is an equilateral  triangle.

Proved that :

i) arc RS arc QS = arc QR

ii) m (arc QRS) = 240°.

​To prove :

( i ) arc RS ≅ arc Qs ≅ arc QR

( ii ) m (arc QRS) = 240°

Given :

From the figure,

Note : Figure has attached below

ΔQRS is an equilateral triangle.

( i ) Finding arc RS = arc QS = arc QR :

An equilateral triangle sides are equal.

So, chord RS = chord QS = chord QR

Corresponding arc of congruent chord of circle are congruent

arc RS = arc QS = arc QR -----> (a)

Hence proved.

( ii ) Finding m (arc QRS) :

Measurement of circle is 360°.

m(arc RS) + m(arc QS) + m(arc QR) = 360° -----> (b)

From the equation (a), it has proven that arc RS = arc QS = arc QR.

So, the equation (b) becomes,

m(arc RS) + m(arc RS) + m(arc RS) = 360°

3 × m(arc RS) = 360°

m(arc RS) = $\frac{360}{3}$ = 120°.

In m(arc QRS),

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

                   = 120° + 120° = 240°

m(arc QRS) = 240°.

Hence proved.

   

To learn more...

1. Prove that the sum of the angles of quadrilateral is 360

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2. State and prove angle sum property of a quadrilateral

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