Solution #32

Consider the following diagram:

PM is the median, and PN is the perpendicular to QR. MR = x, so QM also is x. Letting MN be y, QN is then x-y.

We use the Pythagorean theorem on triangle PNQ to obtain PN = √ 16 - (x-y)²  . We also use the Pythagorean theorem ontriangle PNM to obtain PN = √ (7/2)² - y²  .

Setting these two equal and squaring both sides, we have:

16 - (x-y)² = 49/4 - y²
16 - x² + 2xy - y² = 49/4 - y²
15/4 = x² -2xy

We can also use the Pythagorean theorem on triangle PNR to obtain PN = √ 49 - (x+y)²  . Equating this with the first expression for PN and squaring both sides, we have:

16 - x² + 2xy - y² = 49 - x² - 2xy - y²

33 = 4xy

Substituting in the first expression, we have:

15/4 = x² -33/2
x² = 81/4
x = 9/2

Thus QR = 9.