Point H is the base of the height drawn from the vertex of the right angle B of the triangle ABC

Point H is the base of the height drawn from the vertex of the right angle B of the triangle ABC to the hypotenuse AC. Find AB if AH = 10 and AC = 40

Find the length of the segment CH:
AC = AH + CH;

10 + CH = 40;

CH = 40-10;

CH = 30 conventional units.

The length of the height of a right-angled triangle, drawn to the hypotenuse, can be found by the formula:
BH² = AH * CH.

Substitute the data according to the value condition and find the length of the height:

BH = √ (10 * 30) = √300 (conventional units).

From △ AHB, by the Pythagorean theorem, we find AB (AB is the hypotenuse, AH and BH are legs):
AB² = AH² + BH²;

AB = √ (AH² + BH²);

AB = √ (10² + (√300) ²);

AB = √ (100 + 300);

AB = √400;

AB = 20 conventional units.

Answer: AB = 20 conventional units.