# In trapezoid ABCD the median MN cuts diagonals AC =D at p and q repectively, BC=12, and AD=20 Determine PQ please explain your steps. i know the median is 16 and I know the correct answer is 4, I know there is a formula but we have not proved it.

### Solution:

First, let’s draw the trapezoid so that we can visualize the problem. We have not drawn the trapezoid exactly to scale, but this should be accurate enough for our purposes:

Now, we know that MN = 16 because the median equals the average of the top and bottom of the trapezoid

We also know that the distance of Mp = 6, because it just be half the length of BC (the sides of triangle AMp are half the size of the sides of triangle ABC because the triangles are similar and we know that AM is half the length of AB since MN is the median).Similarly,the length of qN = 6.

Now it’s easy to solve for pq:

- Mp + pq + qN = MN = 16
- 6 + pq + 6 = 16
- pq = 4

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