# One side of the rectangle is 14 cm smaller than the other. Find the sides of a rectangle if its diagonal is 26 cm.

To determine the sides of a rectangle – consider a rectangular

a triangle, the lengths of the legs are equal to the lengths of the sides of the rectangle,

and its hypotenuse is the diagonal of the rectangle.

Let one leg be equal to x, then the second – x – 14. According to the Pythagorean theorem:

x ^ 2 + (x – 14) ^ 2 = 26 ^ 2,

x ^ 2 + x ^ 2 – 2 x 14 + 14 ^ 2 = 26 ^ 2,

2 x ^ 2 – 28 x + 196 = 676,

2 x ^ 2 – 28 x + 196 – 676 = 0,

2 x ^ 2 – 28 x – 480 = 0.

1. Let’s define the discriminant of the equation:

D = (-28) ^ 2 – 4 2 (-480) = 784 + 3840 = 4624.

2. Roots of the equation (values of the 1st leg):

x1 = (28 + v4624) / 2 2 = (28 + 68) / 4 = 96/4 = 24 cm,

x2 = (28 – v 4624) / 2 2 = (28 – 68) / 4 = -40/4 = -10 <0.

3. Calculate the value of the 2nd leg:

x – 14 = 24 – 14 = 10 cm.

Answer: the lengths of the sides of the rectangle are 10 cm and 24 cm.