In a right-angled triangle ABC, angle C = 90 degrees, AB = 8cm, angle ABC = 45 degrees.

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

1. Angle BAC = 180 ° – 45 ° – 90 ° = 45 °.

2. Two angles at the base of AB in a right-angled triangle ABC are equal. Therefore, this triangle is isosceles. Hence, AC = BC.

3. Considering that the ratio AC / AB is the sine of the angle at apex A, we calculate the length of the AC leg:

Sine 45 ° = √2 / 2.

AC / AB = √2 / 2.

AC = AB x √2 / 2 = 8 x √2 / 2 = 4√2 cm.

3. In the BCD triangle, the ratio of the CD leg to the BC hypotenuse is the sine of the 45 ° CBD angle.

СD = ВС x √2 / 2 = 4√2 x √2 / 2 = 4 cm.

Answer: СD = 4 cm, АС = 4√2 cm.