# In a rectangular trapezoid ABCD (AD is parallel to BC, AB is perpendicular to AD), the AC diagonal is perpendicular

In a rectangular trapezoid ABCD (AD is parallel to BC, AB is perpendicular to AD), the AC diagonal is perpendicular to the lateral side of CD, angle D = 30 degrees. find the smaller base of the trapezoid if the larger base is 24cm.

Since, by condition, AC is perpendicular to CD, the ACD triangle is rectangular. The AC leg lies opposite an angle of 30, then its length is equal to half the length of the AD hypotenuse.

AC = AD / 2 = 24/2 = 12 cm.

Angle CAD = 180 – ACD – ADC = 180 – 90 – 30 = 60, then the angle BAC = BAD – CAD = 90 – 60 = 30.

In a right-angled triangle ABC, the leg BC lies opposite the angle 30, then its length is equal to half of the AC.

BC = AC / 2 = 12/2 = 6 cm.

Answer: The length of the smaller base of the trapezoid is 6 cm.

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