The diagonal of the rectangle is 15 cm. If one of its sides is reduced by 6 cm and the other is reduced by 8 cm

The diagonal of the rectangle is 15 cm. If one of its sides is reduced by 6 cm and the other is reduced by 8 cm, then the perimeter will decrease by 3 times. find the sides of the rectangle.

Let us introduce the notation, let the sides of the rectangle be x and y. Two adjacent sides and the diagonal of the rectangle make up a right-angled triangle, according to the Pythagorean theorem:

x² + y² = 15²; x² + y² = 225.

Let us express the perimeter of the rectangle: P = 2 (x + y).

Decrease the sides of the rectangle, the sides will be equal to (x – 6) and (y – 8).

Let us express the perimeter of the resulting rectangle: Pnov = 2 (x – 6 + y – 8) = 2 (x + y – 14).

The resulting perimeter turned out to be 3 times smaller, let’s draw up the equation:

3 * 2 (x + y – 14) = 2 (x + y). Let’s simplify the equation:

6x + 6y – 84 = 2x + 2y.

6x – 2x + 6y – 2y = 84.

4x + 4y = 84. Divide the equation by 4:

x + y = 21.

The result is a system of equations: x² + y² = 225; x + y = 21.

Express x from the second equation and substitute it into the first:

x = 21 – y.

(21 – y) ² + y² = 225.

441 – 42y + y² + y² = 225.

2y² – 42y + 216 = 0.

y² – 21y + 108 = 0.

D = 441 – 432 = 9 (√D = 3);

y1 = (21 – 3) / 2 = 9.

y2 = (21 + 3) / 2 = 12.

x1 = 21 – 9 = 12, x2 = 21 – 12 = 9.

Answer: the sides of the rectangle are 12 cm and 9 cm.