The perimeter of the rectangle is 82 cm. And the length of the diagonal is 29 cm. Find the lengths of the sides of the rectangle?

1. Let one side of the rectangle be x cm, and the other side y cm. From the condition, the sum of the sides of the rectangle, that is, its perimeter is 82 cm. Let’s make an equation and express one side:

P = 2 (x + y);

2 (x + y) = 82;

x + y = 41;

y = 41 – x;

2. Consider a diagonal of 29 cm as the hypotenuse of a right-angled triangle, where x is one leg, and y is its second leg, respectively. Then we use the Pythagorean theorem:

d² = x² + y²;

d = √ (x² + y²);

(x² + (41 – x) ²) = 29²;

(x² + (41 – x) ²) = 841;

x² + 1681 – 82x + x² = 841;

2x² – 82x + 1681 – 841 = 0;

2x² – 82x + 840 = 0;

x² – 41x + 420 = 0;

Find the roots by solving the quadratic equation:

Let’s calculate the discriminant:

D = b² – 4ac = (- 41) ² – 4 * 1 * 420 = 1681 – 1680 = 1;

D ›0 means:

x1 = (- b – √D) / 2a = (41 – √1) / 2 * 1 = (41 – 1) / 2 = 40/2 = 20;

x2 = (- b + √D) / 2a = (41 + √1) / 2 * 1 = (41 + 1) / 2 = 42/2 = 21;

Let’s find the second side:

y = 41 – x

If x1 = 20 cm, then y1 = 41 – 20 = 21 cm;

If x1 = 21 cm, then y1 = 41 – 21 = 20 cm;

Answer: the lengths of the sides of the rectangle are 20 cm and 21 cm, respectively.