The diagonal of an isosceles trapezoid is 20 cm and is perpendicular to the lateral side.

| April 29, 2021 | education

The diagonal of an isosceles trapezoid is 20 cm and is perpendicular to the lateral side. The side and the larger base of the trapezoid are in a ratio of 3: 5. Find the midline of the trapezoid.

Let the length of the lateral side be 3 * X cm, then the larger base is 5 * X cm.

Since AC is perpendicular in CD, the triangle ACD is rectangular, then, by the Pythagorean theorem, AC ^ 2 = AD ^ 2 – CD ^ 2.

400 = 25 * X ^ 2 – 9 * X2 = 16 * X2.

X ^ 2 = 25.

X = 5. Then CD = 3 * 5 = 15 cm, AD = 5 * 5 = 25 cm.

Determine the area of ​​the triangle ACD. Sasd = AC * CD / 2 = 20 * 15/2 = 150 cm2.

Also Sasd = AD * CH / 2.

CH = 2 * Sacd / AD = 300/25 = 12 cm.

In a right-angled triangle CDH, DH ^ 2 = CD ^ 2 – CH ^ 2 = 225 – 144 = 81.

DН = 9 cm.

Then АН = АD – DH = 25 – 9 = 16 cm.

Since the trapezium is isosceles, the height CH divides the base of AD into two segments, the length of the larger of which is equal to the length of the midline of the trapezoid.

KM = AH = 16 cm.

Answer: The middle line of the trapezoid is 16 cm.



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