Comparing again!

Geometry Level pending

In the figure above, the two circles share a center at a point $O$ , and the points $A$ , $B$ and $C$ lie on the outer circle. The radius of the outer circle is $20$ percent greater than the radius of the inner circle, and $AB$ is twice the length of $BC$ . Which one is greater, the perimeter of the triangle or the circumference of the smaller circle?

Circumference of the smaller circle Both are equal Perimeter of the traingle Insufficient information

1 solution

Marta Reece
Jun 20, 2017

$ABC$ is a right triangle, since its hypotenuse is a diameter of the outer circle. Its sides are $2R, R$ and $\sqrt3R$ , so the perimeter is

$R(3+\sqrt3)\approx4.73R$

The relationship between the radii can be written as $R=1.2r$ or $r=\dfrac R{1.2}$ . So the circumference of the smaller circle is

$\pi r^2=\dfrac{\pi}{1.2^2}\approx5.24R$

The circumference of the circle is larger.

Thank you.

Hana Wehbi - 3 years, 11 months ago

Comparing again!

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