The perimeter of an isosceles trapezoid is 124 cm. The smaller base is equal to the side and 20 cm

The perimeter of an isosceles trapezoid is 124 cm. The smaller base is equal to the side and 20 cm less than the other base. Find the area.

Let the length of the smaller base and the side be x cm, then the length of the larger base is x + 20 cm.Knowing that the perimeter is 124 cm, we write the equation

x + x + x + x + 20 = 124;

4x = 124 – 20 = 104;

x = 104/4 = 26 cm.

Therefore, the smaller base and sides are 26 cm, the larger base is 46 cm.

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

m = (26 + 46) / 2 = 72/2 = 36 cm.

In an isosceles trapezoid, the length of the projection of the lateral side onto the larger base is equal to half the difference in the lengths of the bases: (46 – 26) / 2 = 10 cm.

From the right-angled triangle formed by the lateral side, its projection and the height of the trapezoid, according to the Pythagorean theorem, we can find the height:

h ^ 2 = 26 ^ 2 – 10 ^ 2 = 676 – 100 = 576;

h = √576 = 24 cm.

The area of ​​the trapezoid is equal to the product of the midline and the height:

S = h * m = 24 * 36 = 864 cm2.