The circumference of the circle and ...

The circumference of the circle and ratio 3 ratio 4 find the find the ratio of the areas
  • 1 answers

Sia 🤖 3 weeks, 2 days ago

Let r1 and r2 be the radii of two given circles.
Let C1 and C2 be their circumferences respectively.
{tex}{C_1} = 2\pi {r_1}{/tex} and {tex}{C_2} = 2\pi {r_2}{/tex}
{tex} \Rightarrow {/tex} C1 ∶ C= 3 ∶ 4
{tex} \Rightarrow \frac{{{c_1}}}{{{c_2}}} = \frac{3}{4}{/tex}
{tex} \Rightarrow \frac{{2\pi {r_1}}}{{2\pi {r_2}}} = \frac{3}{4}{/tex}
{tex} \Rightarrow \frac{{{r_1}}}{{{r_2}}} = \frac{3}{4}{/tex} ......... (i)
Let A1 and A2 be the areas of the two circles.
{tex} \Rightarrow {A_1} = \pi r_1^2{/tex} and {tex} \Rightarrow {A_2} = \pi r_2^2{/tex}
{tex} \Rightarrow \frac{{{A_1}}}{{{A_2}}} = \frac{{\pi r_1^2}}{{\pi r_2^2}}{/tex}
{tex} = \frac{{r_1^2}}{{r_2^2}}{/tex}
{tex} = {\left( {\frac{{{r_1}}}{{{r_2}}}} \right)^2}{/tex}
{tex} = {\left( {\frac{3}{4}} \right)^2}{/tex} ......... from (i)
{tex} = \frac{9}{{16}}{/tex}
{tex} \Rightarrow {/tex} A: A2 = 9 : 16
Hence, the areas of the two given circles are in the ratio 9: 16.

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