Question

A right triangle is inscribed in a circle as shown. What is the area of the shaded region. Diameter of 12 and 9 cm in triangle

Answer 1

Answer:

$36\pi -18\sqrt{7}$

Step-by-step explanation:

Every inscribed right triangle has its hypotenuse as the same size of the Diameter. Assuming that, this is the case check the picture below.

So the hypotenuse= D=2R=12

1. Shaded Area = Area of the Circle- Area of the Triangle

• Area of Circle:

$A_{circle}=\pi (6)^{2}=36\pi$

• Area of the Triangle:

Finding the Triangle base via the Pythagorean Theorem

$a^2=b^2+c^2\\12^2=9^2+c^2\\144-81=c^2\\c=\sqrt{63}\\c=3\sqrt{7}\approx 7.94$

Now plugging in the base of the triangle on that formula:

$A=\frac{bh}{2}\\A=\frac{3\sqrt{7}*12}{2}\\A=18\sqrt{7}$

Finally,

Shaded Area = $36\pi -18\sqrt{7} cm^{2} \approx 65.48 cm^{2}$