In triangle ABC AB = BC = 5, AC = 6. A circle with center at point O is inscribed in it. Find the area of the triangle AOC.

Let’s build the height BH, which is also the median of the triangle ABC, then AH = CH = AC / 2 = 6/2 = 3 cm.

In a right-angled triangle ABH, according to the Pythagorean theorem, we determine the length of the leg BH.

BH ^ 2 = AB ^ 2 – AH ^ 2 = 25 – 9 = 16.

BH = 4 cm.

Determine the area of the triangle ABC. Savs = BH * AC / 2 = 4 * 6/2 = 12 cm2.

The perimeter of the ABC triangle is equal to: Ravs = 5 + 5 + 6 = 16 cm.

Then the radius of the inscribed circle is: R = OH = 2 * Savs / P = 24/16 = 1.5 cm.

Determine the area of the triangle AOC.

Saos = OH * AC / 2 = 1.5 * 6/2 = 4.5 cm2.

Answer: The area of the AOC triangle is 4.5 cm2.