Given: ABCD is a trapezoid. Diagonal – AC angle BAC = CAD angle AD = 12 ADC angle = 60 degrees Find: perimeter ABCD.

| July 3, 2021 | education

Since the angle BAC = CAD, then AC is the bisector of the acute angle of the trapezoid.

In an isosceles trapezoid, the diagonal is the bisector of an acute angle.

Then, AB = CD = x, angle ABC = 180º – 60º = 120º.

Moreover, the trapezoid can be supplemented to a parallelogram, where AC is the bisector of the acute angle of the parallelogram.

Then, by the property of the bisector of a parallelogram, triangle ABC is isosceles, where AB = BC = x.

Hence, the perimeter ABCD of the trapezoid is

P = AB + BC + CD + AD = 3 * x + 12.

Find x by the cosine theorem from triangles ABC and ADC.

AC ^ 2 = x ^ 2 + x ^ 2 – 2 * x * x * cos (180º – 60) º = 2 * x ^ 2 + x ^ 2 = 3 * x ^ 2.

AC ^ 2 = 12 ^ 2 + x ^ 2 – 2 * 12 * x * cos (60º) = 144 + x ^ 2 – 12 * x.

3 * x ^ 2 = 144 + x ^ 2 – 12 * x.

2 * x ^ 2 + 12 * x – 144 = 0.

x ^ 2 + 6 * x – 72 = 0.

x1,2 = (- 6 ± (36 + 4 * 72) ^ (1/2)) / 2 = (- 6 ± 18) / 2.

x1 = (- 6 + 18) / 2 = 6.

x2 = (- 6 – 18) / 2 = -12 does not fit the meaning of the problem.

Then,

P = 3 * x + 12 = 3 * 6 + 12 = 30.



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