Lines AB and CD, on which the lateral sides of the trapezoid ABCD lie, intersect at point K. AB = 16, BC: AD = 5: 9.

| April 1, 2021 | education

Lines AB and CD, on which the lateral sides of the trapezoid ABCD lie, intersect at point K. AB = 16, BC: AD = 5: 9. Find the length of the segment ВK and the ratio of the areas of the ВKС triangle and the ABCD trapezoid.

Consider triangles ABK and ВСK.

Angle A for triangles is common, angle KAD = KBC as the corresponding angles at the intersection of parallel BC and AD secant AK, then K = BC / AD = 5/9.

Let the length of the segment ВK = X cm, then AK = (16 + X) cm.

ВK / AK = X / (16 + X) = 5/9.

9 * X = 80 + 5 * X.

4 * X = 80.

X = BK = 80/4 = 20 cm.

The areas of such triangles are referred to as the squares of their similarity coefficient.

Skvs / Sakv = 25/81.

Savsd = Savk – Skvs.

Savk = Savsd + Skvs.

Skvs / (Savsd + Skvs) = 25/81.

56 * Skvs = 25 * Savsd.

Skvs / Savsd = 25/56.

Answer: The length of the VC is 20 cm, the area ratio is 25/56.



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