In an isosceles trapezoid, the diagonal divides the acute angle into two equal angles. calculate the ratio of the length of the midline of the trapezoid to the length of its larger base if one of the corners of the trapezoid is 60
The length of the middle line of the trapezoid is: KM = (BC + AD) / 2.
Let us express BC through AD. Since AC is the bisector of the angle BAD, the triangle ABC is isosceles, AB = BC. Then СD = ВС. In a right-angled triangle АDН the angle СDН = 60, then the angle DСН = (90-60) = 30. The leg of DH lies opposite the angle 30, then DH = СD / 2 = BC / 2. Similarly, AK = BC / 2.
Since ВСНK is a rectangle, then KН = ВС. Then AD = AK + KN + DH = (BC / 2 + BC + BC / 2) = 2 * BC.
BC = AD / 2.
Substitute the middle line of the trapezoid into the formula.
KM = (AD / 2 + AD) / 2 = 3 * AD / 4.
KM / AD = 3/4.
Answer: The ratio of the midline to the larger base is 3/4.
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