How to find the sides of a rectangle with a perimeter of 46 and a diagonal of 17 cm.

The two sides of a rectangle and its diagonal form a right-angled triangle. The sides of the rectangle will be the legs, and the diagonal will be the hypotenuse of the right triangle.

The perimeter of a rectangle is equal to the sum of the lengths of its four sides. The sum of the two sides of the rectangle (length and width) is equal to half the perimeter, i.e. 46: 2 = 23 cm

Let one side of the rectangle be x cm, then the second side of the rectangle is (23 – x) cm.For a right-angled triangle formed by the sides of the rectangle and its hypotenuse, we can apply the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs. The sum of the squares of the legs is (x ^ 2 + (23 – x) ^ 2). The square of the hypotenuse is 17 ^ 2. Let’s make an equation and solve it.

x ^ 2 + (23 – x) ^ 2 = 17 ^ 2;

x ^ 2 + 529 – 46x + x ^ 2 = 17 ^ 2;

2x ^ 2 – 46x + 529 = 289;

2x ^ 2 – 46x + 529 – 289 = 0;

2x ^ 2 – 46x + 240 = 0;

x ^ 2 – 23x + 120 = 0;

D = b ^ 2 – 4ac;

D = (- 23) ^ 2 – 4 * 1 * 120 = 529 – 480 = 49; √D = 7;

x = (- b ± √D) / (2a);

x1 = (23 + 7) / 2 = 30/2 = 15 (cm) – the first side of the rectangle;

x2 = (23 – 7) / 2 = 16/2 = 8 (cm) – the first side of the rectangle;

23 – x1 = 23 – 15 = 8 (cm) – the second side of the rectangle;

23 – x2 = 23 – 8 = 15 (cm) – the second side of the rectangle.

Answer. 15 cm, 8 cm.