# The legs of a right-angled triangle are 10 and 24, find the height lowered to the hypotenuse.

In order for us to find the hypotenuse of the AC, we will use the Pythagorean theorem which is applicable in a right-angled triangle. The Pythagorean theorem sounds like this: the square of the hypotenuse is equal to the sum of the squares of the legs.

c ^ 2 = a ^ 2 + b ^ 2, where a and b are the legs of a right triangle, and c is the hypotenuse in a right triangle.

In our case, the theorem will look like this: AC ^ 2 = AB ^ 2 + BC ^ 2. Let’s substitute the known values into this formula such as the AB leg which is 10 cm and the second BC leg which is 24 cm and find the AC hypotenuse, then we get:

AC ^ 2 = AB ^ 2 + BC ^ 2;

AC ^ 2 = (10 cm) ^ 2 + BC ^ 2;

AC ^ 2 = (10 cm) ^ 2 + (24 cm) ^ 2;

AC ^ 2 = 100 cm2 + 576 cm2;

AC ^ 2 = 676 cm2;

Find an AC without a square, we get:

AC = √676 cm2 = 26 cm.

And so we found the hypotenuse AC.

Find the projection of the legs on the hypotenuse AH and HC

And so for this we use the formula and find the projections AH and HС:

AB = √AC * AH;

10 cm = √26 cm * AH;

AH = 100 cm2 / 26 cm = 50/13 cm.

BC = √AC * CH;

24 cm = √26 cm * CH;

CH = 576 cm2 / 26 cm = 288/13 cm.

Find the height BH

We use the formula to find the height in a right-angled triangle through the projection of the legs to the hypotenuse, we get:

BH = √ CH * BH = √ 50/13 cm * 288/13 cm = 120/13 cm

Answer: BH = 120/13 cm.