A right-angled triangle is inscribed in a circle with a radius of 5 cm and is described

A right-angled triangle is inscribed in a circle with a radius of 5 cm and is described around a circle with a radius of 1 cm. Find the area of the triangle?

Let us determine what the hypotenuse of the triangle will be equal to (it is the diameter of the circumscribed circle), when it is known from the condition that the radius of the circumscribed circle is 5 cm:

5 * 2 = 10.

As we know from the school curriculum, the radius of the inscribed circle can be determined by the formula:

(a + b – c): 2.

Then we have the expression:

(a + b – 10): 2 = 1;

a + b = 12;

We already know that a ^ 2 + b ^ 2 = 10 ^ 2 = 100.

Therefore, let’s square the first expression:

a ^ 2 + 2ab + b ^ 2 = 144.

Let’s substitute the second into it:

2ab + 100 = 144;

2ab = 44.

The area of the triangle is 1 / 2ab. That is, we get:

44: 4 = 11.

Answer: 11 cm2.