In an isosceles triangle ABC; AB = BC = 13; AC = 10. O is the point of intersection of the bisectors.
In an isosceles triangle ABC; AB = BC = 13; AC = 10. O is the point of intersection of the bisectors. Find the area of the triangle AOC.
Determine the area of the triangle ABC.
The bisector BH, lowered to the base of the AC of an isosceles triangle, is also the height and median of the triangle. Then AH = CH = AC / 2 = 10/2 = 5 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 = 169 – 25 = 144.
BH = 12 cm.
Determine the area of the triangle ABC. Savs = AC * BH / 2 = 10 * 12/2 = 60 cm2.
The point O of intersection of the bisectors is the center of the inscribed circle, and its radius OH is the height of the AOC triangle.
Determine the radius of the inscribed circle. R = OH = 2 * Savs / (AB + BC + AC) = 2 * 60 / (13 + 13 + 10) = 120/36 = 10/3 cm.
Determine the area of the triangle AOC.
Saos = AC * OH / 2 = (10 * 10/3) / 2 = 100/6 = 50/3 = 16 (2/3) cm2.
Answer: The area of the AOC triangle is 16 (2/3) cm2.
