The diagonal of an isosceles trapezoid, the perimeter of which is 52, divides the obtuse angle of the trapezoid

| July 1, 2021 | education

The diagonal of an isosceles trapezoid, the perimeter of which is 52, divides the obtuse angle of the trapezoid in half. The smaller base of the trapezoid is 1. Find the area of the trapezoid.

By condition, AC is the diagonal and bisector of angle C, then the angle ACB = ACD.

Angle CAD = ACB as cross-lying angles at the intersection of straight lines AD and BC secant AC, then angle ACD = CAD, and triangle ACD is isosceles and segment AD = CD.

Then AB = AD = CD.

Let the side AB = X cm, then the perimeter of the trapezoid is:

P = AB + BC + CD + AD = 3 * X + 1 = 52 cm.

3 * X = 52 – 1 = 51 cm.

X = 51/3 = 17 cm.

AB = CD = AD = 17 cm.

Let us lower the height CH from the top C.

Segment DH = (AD – BC) / 2 = (17 – 1) / 2 = 8 cm.

From the right-angled triangle НDС we determine the height of the trapezoid СН according to the Pythagorean theorem.

CH ^ 2 = CD ^ 2 – DH ^ 2 = 17 ^ 2 – 8 ^ 2 = 289 – 64 = 225.

CH = 15 cm.

Determine the area of ​​the trapezoid.

S = (AD + BC) * CH / 2 = (17 + 1) * 15/2 = 135 cm2.

Answer: The area of ​​the trapezoid is 135 cm2.



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