In a right-angled triangle ABC, angle C is 90 degrees, AB = 8 cm, angle ABC

In a right-angled triangle ABC, angle C is 90 degrees, AB = 8 cm, angle ABC is 45 degrees. Find: a) AC; b) the height of CD, drawn to the hypotenuse.

1. We calculate the value of the angle at the vertex A:

Angle A = 180 ° – 45 ° – 90 ° = 45 °.

2. Angle A = angle B = 45 °. Therefore, triangle ABC is isosceles. Hence, AC = BC.

3. We calculate the length of the legs AC and BC, applying the formula of the Pythagorean theorem:

AB ^ 2 = AC ^ 2 + BC ^ 2 = 2AC ^ 2.

AC ^ 2 = AB ^ 2/2 = 64/2.

AC = √64 / 2 = 4√2 centimeters.

4. We calculate the length of the height CD through the sine of the angle A:

CD / AC = sine 45 ° = √2 / 2.

CD = AC x √2 / 2 = 4√2 x √2 / 2 = 4 centimeters.

Answer: AC = 4√2 centimeters, CD = 4 centimeters.