# Shaded Area – Right Triangle Inscribed in a Circle

A right triangle is inscribed in a circle such that its longest side is the diameter of the circle. If the shorter sides of the triangle measure 6cm and 8cm, find the area of the shaded region. Use $\pi = 3.14$

Solution

A triangle inscribed in a circle with its longest side as the diameter of the circle is always a right triangle (by Thales’ Theorem). So, we can find the area $A_T$ of the triangle.

Finding the area of the triangle,

$A_T = \frac{6 \times 8}{2} = 24$

By the Pythagorean Theorem, the square of the longest side $c$ is equal to the sum of the shorter sides $a$ and $b$. That is,

$c^2 = a^2 + b^2$
$c^2 = 6^2 + 8^2$
$c^2 = 36 + 64$
$c^2 = 100$
$c = 10$.

So, the radius of the circle is equal to 10/2 = 5.

Therefore, the area of the circle is

$\pi r^2 = (\pi)(5^2) = (25) (3.14) = 78.5$

Now, to get the area of the shaded region, we subtract the area of the triangle from the area of the circle.

$A_S = A_C - A_T$

$A_S = 78.5 - 24 = 54.5$

The correct answer is $54.5$ square centimeters.

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