In the triangle ABC AB = 13, BC = 14, AC = 15. AH- height. Find BH and CH.

By Heron’s theorem, we determine the area of the triangle ABC.

The semi-perimeter of the triangle is: p = (AB + BC + AC) / 2 = (13 + 14 + 15) / 2 = 21 cm.

Then Sav = √21 * (21 – 13) * (21 – 14) * (21 – 15) = √ 7056 = 84 cm2.

Also, the area of the triangle ABC is equal to: Savs = BC * AH / 2.

84 = 14 * AH / 2.

AH = 2 * 84/14 = 12 cm.

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

BH ^ 2 = AH ^ 2 – AB ^ 2 = 169 – 144 = 25.

BH = 5 cm.

Then the length of the segment CH = CB + BH = 14 + 5 = 19 cm.

Answer: The length of the segment BH is 5 cm, CH is 19 cm.