In a right-angled triangle, the angle is 45, and the height drawn to the hypotenuse is 9 cm. Find the area of this triangle.

Let ABC be a given triangle (angle C is 90 °), angle A is 45 °, CH is height, CH = 9 cm.

The sum of the angles in a triangle is 180 °. We calculate the value of the angle B: 180 ° – (90 ° + 45 °) = 45 °.

Hence, triangle ABC is isosceles, AC = BC. This means that the CH height is also the median.

The median of a right-angled triangle, drawn from a right angle, is half the hypotenuse.

This means that the hypotenuse AB = 9 * 2 = 18 cm.

The area of a triangle is half the product of a side by the height drawn to that side.

S = 1/2 * AB * CH = 1/2 * 18 * 9 = 81 (cm²).