A geometry problem by Hana Wehbi

Geometry Level 3

Three circles, each with a radius of 10 cm, are drawn tangent to each other so that their centres are all in a straight line. These circles are inscribed in a rectangle which is inscribed in another circle. The area of the largest circle is?

Problem and image: Courtesy Waterloo University
100 π 100\pi 800 π 800\pi 1000 π 1000\pi 900 π 900\pi 2000 π 2000\pi

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

By Pythagorean Theorem ,

r = 3 0 2 + 1 0 2 = 1000 r = \sqrt{30^2+10^2}=\sqrt{1000}

Solving for the area, we have

A = π r 2 = π ( 1000 ) 2 = A = \pi r^2 = \pi (\sqrt{1000})^2 = 1000 π \boxed{\color{#D61F06}\large1000 \pi}

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Thank you.

Hana Wehbi - 4 years, 2 months ago
Hana Wehbi
Apr 3, 2017

r 2 = 3 0 2 + 1 0 2 = 1000 r^2=30^2+10^2=1000

A = π r 2 = π ( 1000 ) = 1000 π A=\pi r^2= \pi(1000)=1000\pi

By symmetry, the center of the large circle is the center of the smaller middle circle.

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