Question

Fill In The Blanks

Two congruent circles have centres at O and O'. Arc AXb of circle centred at O, subtends an angle of 75^∘ at the centre O and arc PYQ ( or circle centred at O') subtends an angle 25^∘ at the centre O'. The ratio of the arcs AXB and PYQ is ___________.

Answer 1

Given:
O and O' are the centres of two congruent circles
AXB of circle centred at O, subtends an angle of 75^∘ at the centre O
arc PYQ of circle centred at O' ,subtends an angle 25^∘at the centre O'



Since, the circles are congruent
Therefore, they have same radius of measure r cm. ...(1)

We know, Length of arc = $\frac{\theta }{360°}\times 2\mathrm{\pi }r$

Thus,
$$\begin{gathered} \mathrm{Length}\mathrm{of}\mathrm{arc}AXB=\frac{75°}{360°}\times 2\mathrm{\pi }r...\left(2\right) \\ \mathrm{Length}\mathrm{of}\mathrm{arc}PYQ=\frac{25°}{360°}\times 2\mathrm{\pi }r...\left(3\right) \\ \frac{\mathrm{Length}\mathrm{of}\mathrm{arc}AXB}{\mathrm{Length}\mathrm{of}\mathrm{arc}PYQ}=\frac{{\displaystyle \frac{75°}{360°}}\times 2\mathrm{\pi }r}{{\displaystyle \frac{25°}{360°}}\times 2\mathrm{\pi }r} \\ =\frac{{\displaystyle 75°}}{{\displaystyle 25°}} \\ =\frac{{\displaystyle 3}}{{\displaystyle 1}} \end{gathered}$$


Hence, the ratio of the arcs AXB and PYQ is 3 : 1.