In triangle ABC, angle C is 90 degrees, CH is height, AB = 16, sine A = 3/4. Find AH.

Given:

ABC – right triangle;

Angle C = 90 degrees;

CH – height;

AB = 16;

sin A = 3/4;

Find AH.

Decision:

1) sin a = BC / AB;

From here we get that BC = AB * sin a = 16 * ¾ = 16/4 * 3 = 4 * 3 = 12;

2) Find the AC from the Pythagorean theorem.

AC = √ (AB ^ 2 – BC ^ 2) = √ (16 ^ 2 – 12 ^ 2) = √ (256 – 144) = √112 = √ (16 * 7) = √16 * √7 = 4 * √ 7;

2) From the formula AC ^ 2 = AB * AH we find AH.

In order to find the hypotenuse AB of the triangle ABC, we use the formula:

AH = AC ^ 2 / AB;

Substitute the known values into the formula and find AH.

AH = AC ^ 2 / AB = (4 * √7) ^ 2/16 = 16 * 7/16 = 16/16 * 7 = 1 * 7 = 7;

Answer: AH = 7.