The leg of a right-angled triangle is 12 cm, and its projection onto the hypotenuse is 8 cm.

The leg of a right-angled triangle is 12 cm, and its projection onto the hypotenuse is 8 cm. Find the area of the triangle.

Let ABC be a right-angled triangle given by condition, AB and AC = 12 cm – legs, BC – hypotenuse.
Let us draw the vertex AH from the vertex A to the hypotenuse. The BH segment is the projection of the AB leg onto the hypotenuse, and the HC = 8 cm segment is the projection of the AC leg onto the hypotenuse.
Consider the triangle AHC: AC = 12 cm – hypotenuse (since it lies opposite the angle AHC, which is 90 degrees, since AH is the height, that is, the perpendicular lowered to BC), HC = 8 cm is the leg.
Each leg of the triangle is the geometric mean of the hypotenuse and the projection of the leg onto the hypotenuse, that is:
AC ^ 2 = ВС * НС;
12 ^ 2 = BC * 8;
8BC = 144;
BC = 18 cm.
In triangle ABC, the hypotenuse BC = 18 cm, leg AC = 12 cm are known.Let’s find the second leg AB according to the Pythagorean theorem:
AB = √ (BC ^ 2 – AC ^ 2);
AB = √ (18 ^ 2 – 12 ^ 2) = √ (324 – 144) = √180 = 6√5 (cm).
The area of ​​the triangle ABC is equal to half the product of its legs:
S = (AB * AC) / 2;
S = (6√5 * 12) / 2 = 36√5 (cm square).
Answer: S = 36√5 cm square.