In an isosceles trapezoid, one base is 4 times the size of the other. The perimeter of the trapezoid is 20 cm

| April 10, 2021 | education

In an isosceles trapezoid, one base is 4 times the size of the other. The perimeter of the trapezoid is 20 cm, the height is 4 cm. Find the sides of the trapezoid.

Let the length of the base BC = X cm, and the length of the lateral side is equal to Y cm.

Then AD = 4 * BC = 4 * X cm.

Let’s draw the height of the DK, which divides the base AD into two segments, the length of the smaller of which is equal to: AK = (AD – BC) / 2 = (4 * X – X) / 2 = 3 * X / 2 cm.

In a right-angled triangle ADK,

AB ^ 2 = VK ^ 2 + AK ^ 2.

Y2 = 16 + 9 * X ^ 2/4.

4 * Y2 = 64 + 9 * X ^ 2 (1).

The perimeter of the trapezoid is: P = (AB + BC + CD + AD) = 5 * X + 2 * Y = 20 cm.

2 * Y = 20 – 5 * H.

4 * Y ^ 2 = (20 – 5 * X) ^ 2 = 400 – 200 * X + 25 * X ^ 2 (2).

The left sides of equations 1 and 2 are equal, then:

64 + 9 X ^ 2 = 400 200 * X + 25 * X ^ 2.

16 * X ^ 2 – 200 * X + 336 = 2 * X2 – 25 * X + 42 = 0

Let’s solve the quadratic equation.

X1 = BC = 2 cm.

X2 = 10.5 cm. (Does not fit because P = 20 cm.)

AD = 2 * 4 = 8 cm.

AK = 3 * 2/2 = 3 cm.

AB ^ 2 = 16 + 9 = 25.

AB = CD = 5 cm.

Answer: The sides of the trapezoid are equal: 5 cm, 2 cm, 5 cm, 8 cm.



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