Find the side AB of trapezoid ABCD if the angles ABC and BCD are 30 ° and 120 °, respectively, and CD = 25.

| January 15, 2021 | education

Let a trapezoid ABCD be given, for which the angles ABC and BCD are equal to 30 ° and 120 °, respectively, and CD = 25. From the obtuse angle BCD we draw the height СН, which cuts off the rectangular Δ НCD, in which ∠НCD = 120 ° – 90 ° = 30 °. By the property of the leg, which lies opposite an angle of 30 °, we obtain HD = CD: 2; НD = 25: 2 = 12.5. The height of the trapezoid НC is found from Δ НCD by the Pythagorean theorem: CD² = НD² + НC²; 25² = 12.5² + HC²; HC = 12.5 ∙ √3. From the top BAD of trapezoid ABCD we draw the height of AK, which will cut off the rectangular Δ AKB, in which the leg AK = HC = 12.5 ∙ √3 lies opposite the angle ∠ABC = 30 °, then the hypotenuse AB = 2 ∙ AK; AB = 2 ∙ 12.5 ∙ √3 = 25 ∙ √3. And this is the side AB of the trapezoid ABCD.
Answer: The side AB of trapezoid ABCD is 25 ∙ √3.



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