A perpendicular is dropped from the top of the rectangle to the diagonal, which divides it into

A perpendicular is dropped from the top of the rectangle to the diagonal, which divides it into segments 9 cm and 16 cm long. Find the S of the rectangle.

1. Vertices of the rectangle A, B, C, D. BK – perpendicular to the AC diagonal.

AK = 9 centimeters. CK = 16 centimeters.

2. Since the height BK is drawn from the vertex of the right angle of the triangle ABC, its length is calculated by the formula:

BK = √AK x CK = √9 x 16 = √144 = 12 centimeters.

3. The side of the rectangle AB is the hypotenuse of the right-angled triangle ABK.

We calculate its length using the Pythagorean theorem:

AB = √AK² + BK² = √9² + 12² = √81 + 144 = √225 = 15 centimeters.

4. The side of the rectangle BC is the hypotenuse of the rectangular triangle BC.

We calculate its length, also using the Pythagorean theorem:

BC = √CK² + BK² = √16² + 12² = √256 + 144 = √400 = 20 centimeters.

5. Calculate the area (S) of a given rectangle:

S = AB x BC = 15 x 20 = 300 centimeters².

Answer: the area of ​​a given rectangle is 300 centimeters².