The leg of a right-angled triangle is 30 cm, and its projection onto the hypotenuse is 18 cm. Find the hypotenuse.

1. Let’s designate the vertices of a right-angled triangle ABC, angle B = 90 °. BH is the height drawn from vertex B to the hypotenuse AC.

2. We calculate the length BH using the Pythagorean theorem:

BH = √AB ^ 2 – AH ^ 2 = √30 ^ 2 – 18 ^ 2 = √900 – 324 = √576 = 24 cm.

3. Using the formula for calculating the height, drawn from the vertex of the right angle to the hypotenuse, we calculate the length of the CH segment:

BH ^ 2 = AH x CH;

CH = BH ^ 2 / AH = 24 ^ 2/18 = 576/18 = 32 cm.

4. Calculate the length of the AC hypotenuse:

32 + 18 = 50 cm.

Answer: the length of the AC hypotenuse is 50 cm.