The sides of the triangle are 9 and 24, and the angle between them is 90 degrees. Find the perimeter and area.

This triangle is rectangular. The sides AB and AC, which are 9 cm and 24 cm, are legs, since they are adjacent to a right angle. Using the Pythagorean theorem, we find the length of the BC hypotenuse:

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

BC ^ 2 = 9 ^ 2 + 24 ^ 2 = 81 + 576 = 657;

BC = √657 = 25.6 cm.

We are looking for the perimeter of the triangle:

P = AB + BC + AC;

P = 9 + 24 + 25.6 = 58.6 cm.

Now we find the area of the triangle using Heron’s formula:

S = √p (p-a) (p-b) (p-c);

p = (a + b + c) / 2;

p = (9 + 24 + 25.6) / 2 = 58.6 / 2 = 29.3 cm;

S = √29.3 (29.3 – 9) (29.3 – 24) (29.3 – 25.6) = √29.3 20.3 5.3 3.7 = √11663.8 ≈ 108 cm2.

Answer: the perimeter of the triangle is 58.6 cm, the area is 108 cm2.