Geometry

Question

0

7

1

+821

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

CPhill · Answer 1 · 2025-01-12T08:06:37

P

5 8

Q M R

5.5 5.5

Note that cos PMR = -cos PMQ

Law of Cosines

PQ^2 = PM^2 + QM^2 - 2 ( PM * QM) cos (PMQ)

PR^2^2 = PM^2 + RM^2 - 2(PM * RM) cos (PMR)

5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)

8^2 = PM^2 + 5.5^2 - 2(PM * 5.5) (- cos PMQ)

5^2 = PM^2 + 5.5^2 - 2 (PM * QM) cos (PMQ)

8^2 = PM^2 + 5.5^2 + 2(PM * 5.5) ( cos PMQ) add these

89 = 2PM^2 + 60.5

[89 - 60.5] / 2 = PM^2

PM = sqrt [ 28.5 / 2 ] ≈ 3.775