In right-angled triangle ABC, angle C is straight, CH-height. Find AC if AH = 2, AB = 32.

In a right-angled triangle ABC it is known:

Angle C = 90 °;
CH – height;
AH = 2;
Hypotenuse AB = 32.
Find the AC leg.

Solution:

1) Find the side BH.

BH = AB – AH = 32 – 2 = 30;

2) CH ^ 2 = AH * BH;

CH = √ (AH * BH);

Substitute the known values into the formula.

CH = √ (2 * 30) = √60 = √ (4 * 15) = √4 * √15 = 2 * √15 = 2√15;

3) Consider a right-angled triangle ACH with a right angle H.

AC = √ (AH ^ 2 + CH ^ 2) = √ (2 ^ 2 + (2√15) ^ 2) = √ (4 + 4 * 15) = √ (4 + 60) = √64 = √8 ^ 2 = 8;

This means that the leg AC of the triangle ABC is equal to AC = 8.