The height of a right-angled triangle ABC, drawn from the top of the right angle, divides the hypotenuse

The height of a right-angled triangle ABC, drawn from the top of the right angle, divides the hypotenuse into segments equal to 4cm and 9cm. Find the area of this triangle.

1. A, B, C – the vertices of the triangle. ∠С = 90 °. CE – height. AE = 4 cm. BE = 9 cm. S is the area of the triangle.

2. According to the properties of a right-angled triangle, the height CE, drawn from the vertex of the right angle, is calculated by the formula:

CE = √AE x BE = √4 x 9 = 6 cm.

3. ВС = √CE² + BE² (by the Pythagorean theorem).

BC = √6² + 9² = √36 + 81 = √117 = 3√13 cm.

4. AC = √CE² + AE² = √6² + 4² = √36 + 16 = √52 = 2√13 cm.

5. S triangle = AC x BC / 2 = 2√13 x 3√13 / 2 = 39 cm².

Answer: S of a triangle is 39 cm².