Question

Given: ∆ABC, m∠C = 90° CB = 8, m∠B = 38º Find the area of a circumscribed circle. Find the area of the inscribed circle.

Answer 1

Answer:

Circumscribed circle: Around 80.95

Inscribed circle: Around 3.298

Step-by-step explanation:

Since C is a right angle, when the circle is circumscribed it will be an inscribed angle with a corresponding arc length of 2*90=180 degrees. This means that AB is the diameter of the circle. Since the cosine of an angle in a right triangle is equivalent to the length of the adjacent side divided by the length of the hypotenuse:

$\cos 38= \dfrac{8}{AB} \\\\\\AB=\dfrac{8}{\cos 38}\approx 10.152$

To find the area of the circumscribed circle:

$r=\dfrac{AB}{2}\approx 5.076 \\\\\\A=\pi r^2\approx 80.95$

To find the area of the inscribed circle, you need the length of AC, which you can find with the Pythagorean Theorem:

$AC=\sqrt{10.152^2-8^2}\approx 6.25$

The area of the triangle is:

$A=\dfrac{bh}{2}=\dfrac{8\cdot 6.25}{2}=25$

The semiperimeter of the triangle is:

$\dfrac{10.152+6.25+8}{2}\approx 24.4$

The radius of the circle is therefore $\dfrac{25}{24.4}\approx 1.025$

The area of the inscribed circle then is $\pi\cdot (1.025)^2\approx 3.298$.

Hope this helps!