In trapezoid ABCD with bases AD = 10 and BC = 5, the line passing through point A

| February 14, 2021 | education

In trapezoid ABCD with bases AD = 10 and BC = 5, the line passing through point A and the midpoint of the diagonal BD intersects side CD at point L and line BC at point K. Find LD if CD = 9.

Consider the triangles KВO and ADO, which are similar in two angles, the angle of the ВOK and AOD are equal as vertical, the angle of the EKР is equal to the KAD as crosswise.

Since, by condition, ВO = OD, the triangles ВOK and AOD are equal in side and two adjacent angles.

Then ВK = AD = 10 cm, СK = ВK – ВС = 10 – 5 = 5 cm.

Triangles ALD and CLK are similar in two angles.

Let the length of the segment CL = X cm, then DL = (9 – X) cm.

СK / BP = CL / DL.

5/10 = X / (9 – X).

10 * X = 45 – 5 * X.

15 * X = 45.

X = CL = 45/15 = 3 cm.

LD = 9 – 3 = 6 cm.

Answer: The length of the segment LD is 6 cm.



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