In an isosceles trapezoid, the larger base is 25, and the diagonal is perpendicular to the lateral side.

| May 29, 2021 | education

In an isosceles trapezoid, the larger base is 25, and the diagonal is perpendicular to the lateral side. Find the area of a trapezoid if its side is 1.25 times its height.

Let’s draw the height BH of the trapezoid АBСD. Let the length of the height BH = X cm, then AB = 1.25 * X cm.

Determine the sine of the angle BAН.

SinBАН = BН / АН = X / 1.25 * X = 0.8. Since the trapezoid ABCD is isosceles, then SinADC = 0.8.

Triangle ACD is rectangular by condition, then AC = AD * SinADC = 25 * 0.8 = 20 cm.

Then CD ^ 2 = AD ^ 2 – AC ^ 2 = 625 = 400 = 225. CD = AB = 15 cm.

Since AB = 1.25 * BH, then BH = 15 / 1.25 = 12 cm.

AH ^ 2 = AB ^ 2 – BH ^ 2 = 225 – 144 = 81. AH = 9 cm.

In an isosceles trapezoid: AH = (AD – BC) / 2.

Then BC = AD – 2 * AH = 25 – 18 = 7 cm.

Determine the area of the trapezoid.

Savsd = (АD + ВС) * BН / 2 = (25 + 7) * 12/2 = 192 cm2.

Answer: The area of the trapezoid is 192 cm2.



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