Find the length of the side of an isosceles trapezoid if it is equal to its midline and the perimeter of the trapezoid is 24 cm.

Given an isosceles trapezoid ABCD: AB = CD.

The middle line of a trapezoid is half the sum of the bases:

m = (a + b) / 2.

Thus:

AB = CD = (AD + BC) / 2.

The perimeter of the polygon is equal to the sum of the lengths of all its sides, then the perimeter of the trapezoid ABCD is:

P = AB + BC + CD + AD.

By condition, the perimeter of the trapezoid ABCD is 24 cm, then:

AB + BC + CD + AD = 24.

Let’s replace and substitute (AD + BC) / 2 instead of AB and CD:

(AD + BC) / 2 + BC + (AD + BC) / 2 + AD = 24;

(AD + BC + AD + BC) / 2 + (2 * BC) / 2 + (2 * AD) / 2 = 24;

(2 * AD + 2 * BC) / 2 + (2 * AD + 2 * BC) / 2 = 24;

(2 * AD + 2 * BC + 2 * AD + 2 * BC) / 2 = 24;

(4 * AD + 4 * BC) / 2 = 24;

4 * AD + 4 * BC = 48 (proportional);

AD + BC = 12 cm.

Substitute this value into the expression for the side length:

AB = CD = (AD + BC) / 2 = 12/2 = 6 (cm).

Answer: AB = CD = 6 cm.