In a right-angled triangle ABC from the vertex C of the right angle, the height CH is drawn.

In a right-angled triangle ABC from the vertex C of the right angle, the height CH is drawn. Find the area of a triangle if AC = 13, CH = 12.

In the right-angled triangle ACH we find the leg AH using the Pythagorean theorem.
AH = √ (AC² – CH²) = √ (169 – 144 = √25 = 5
Using the property of the height drawn from the right angle to the hypotenuse, we find BH.
CH² = AH * BH
BH = CH² / AH = 144/5 = 28.8

In the triangle ABC we find the hypotenuse AB and the leg BC:
AB = AH + BH = 5 + 28.8 = 33.8
ВС = √ (AB² – AC²) = √ (1142.44 – 169 = √973.44 = 31.2
Now we find the area of the triangle ABC:
S = 1/2 * a * b = 1/2 * AC * BC = 1/2 * 13 * 31.2 = 202.8
Answer: the area of the triangle ABC is 202.8.