Question

Answer the following :

ΔPQR is an equilateral triangle with side 18 cm. A circle is drawn on segment QR as diameter. Find the length of the arc of this circle within the triangle.

Answer 1

Let O be the centre of the circle drawn on QR as a diameter.

Let the circle intersect seg PQ and seg PR at points M and N respectively.

Since, ${\displaystyle l\left(\text{OQ}\right)=l\left(\text{OM}\right)}$,

∴ m∠OMQ = m∠OQM = 60°

∴ m∠MOQ = 60°

Similarly, m∠NOR = 60°

Given, QR = 18 cm

∴ r = 9 cm

∴ θ = 60°

= ${\displaystyle {(60\times \frac{\pi }{180})}^{\text{c}}}$

= ${\displaystyle \frac{{\pi }^{\text{c}}}{3}}$

∴ length of the arc MN = S = rθ

= ${\displaystyle 9\times \frac{\pi }{3}}$

= 3π cm