Area

Geometry Level 2

A circle passes through three points $P$ , $Q$ and $R$ . Given that $PR$ is a diameter of circle, $PQ$ is 24 and $QR$ is 7, find the area of the shaded region.

Details and Assumptions :

Use the approximation $\pi=3.14$ .

145.6 201.5 190.8 161.3

1 solution

A Former Brilliant Member
Oct 5, 2017

By Thales' theorem, $PQR$ is a right triangle. Consequently, $PR^2=PQ^2+QR^2$ and $PR=25$ .

Now, using Heron's formula we can determine the area of triangle $PQR$ : $A=\sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is the semiperimeter. $A=84$ .

Subtracting A from the area of the semicircle gives the area of the shaded region : $\frac{3.14*12.5^2}{2}-84 \simeq 161.3$

You don't need to use heron's formula as the triangle is a right angle triangle it's base and height will be $7$ and $24$ you can find are from $\frac{1}{2}base\times height$ formula itself

Zakir Husain - 12 months ago

Area

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