In the trapezoid ABCD with bases AD = 32 cm and BC = 18 cm, a middle line PQ is drawn, which intersects the diagonals

| September 20, 2021 | education

In the trapezoid ABCD with bases AD = 32 cm and BC = 18 cm, a middle line PQ is drawn, which intersects the diagonals AC and BD at points M and N. Determine the size of the segment MN.

Let’s take advantage of the fact that the middle line of the trapezoid is parallel to the bases BC and AD.
consider a triangle DBC.

Since point Q is the midpoint of side CD and line NQ is parallel to BC, then

NQ is the midline of triangle DBC. Hence,

Since triangles DNQ and DBC are similar in three angles, then

NQ / BC = DQ / CD = 1/2,

NQ = 1/2 * BC = 1/2 * 18 = 9.

Similarly, consider the triangle ACD.

Since point Q is the midpoint of side CD and line NM is parallel to AD, then

MQ is the midline of the ACD triangle. Hence,

MQ = 1/2 * AD = 1/2 * 32 = 16.

As a result, we get

MN = MQ – NQ = 16 – 9 = 7.



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