# A trapezoid ABCK is given, AK is a larger base, the lateral sides are continued until the intersection at point M.

A trapezoid ABCK is given, AK is a larger base, the lateral sides are continued until the intersection at point M. Prove that the triangles AMK and BMC are similar. Find the base BC if BM = 8, AB = 4, AK = 18

In triangles AMK and BMC, the angle M is common, the angle MAK = MBC as the corresponding angles at the intersection of parallel lines BC and AK secant AM, then the triangles AMK and BMC are similar in two angles, which was required to be proved.

The length of the segment is AM = BM + AB = 8 + 4 = 12 cm.

Let’s determine the coefficient of similarity of triangles. K = BM / AM = 8/12 = 2/3.

Then BC / AK = 2/3.

BC = 2 * AK / 3 = 2 * 18/3 = 12 cm.

Answer: The base length of the BC is 12 cm.

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