In the trapezoid ABCD, the extensions of the lateral sides AB and CD intersect at point P, Q is the intersection point

| March 8, 2021 | education

In the trapezoid ABCD, the extensions of the lateral sides AB and CD intersect at point P, Q is the intersection point of the diagonals of this trapezoid. Find the ratio of the length of the smaller of the bases of this trapezoid to the length of the larger of the bases, if it is known that the area of triangle ABQ is 1/4 of the area of the triangle ACP.

By the property of a trapezoid, the midpoints of the bases of the trapezoid, the point of intersection of the diagonals, and the point of intersection of the lateral sides lie on one straight line.

The area of ​​the triangle is Sars = Sawq + Svrq + Spcq.

Svрq = Spcq since РQ is the median of triangles BPQ and BCQ, then:

Sars = Sawq + 2 * Svrq.

By condition, Sars = 4 * Sawq.

4 * Savq = Savq + 2 * Svrq.

3 * Savq = 2 * Svrq.

Svrq / Sawq = 3/2.

Triangles BPQ and ABQ have the same height, then Svrq / Sawq = BP / AB = 3/2.

3 * AB = 2 * BP.

AB = 2 * BP / 3.

AP = BP + 2 * BP / 3 = 5 * BP / 3.

BP / AR = 3/5.

Triangles APD and VRS are similar in two angles, then: BC / AD = BP / AP = 3/5.

Answer: The base ratio is 3/5.



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