# Find the sides of the rectangle if their difference is 23 and the diagonal of the rectangle is 37.

March 25, 2021 | education

| To solve the problem, consider the figure.

By condition:

a – b = 23.

a = 23 + b.

AC = 37.

By the Pythagorean theorem, AC ^ 2 = a ^ 2 + b ^ 2.

37 ^ 2 = (23 + b) ^ 2 + b ^ 2.

1369 = 529 + 46 * b + 2 * b ^ 2.

2 * b ^ 2 + 46 * b – 840 = 0.

b ^ 2 + 23 * b – 420 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 23 ^ 2 – 4 * 1 * (-420) = 529 + 1680 = 2209.

b1 = (-23 – √2209) / (2 * 1) = (-23 – 47) / 2 = -70 / 2 = -35. (does not match, since <0).

b2 = (-23 + √2209) / (2 * 1) = (-23 + 47) / 2 = 24/2 = 12.

Side b = 12 cm.

a = 23 + 12 = 32 cm.

Answer: The sides of the rectangle are 12 and 32 cm.

One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.