Question

The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches and $7$ inches. The median $PM$ is $3\dfrac12$ inches. Then $QR,$ in inches, is:

$6$

$7$

$8$

$9$

$10$

Answer: D

Concepts:median (geometry)Stewart’s Theorem law of cosines

Difficulty rating: 1210

Small Hint:

Use Apollonius’s theorem for the median to $QR$

Big Hint:

Substitute $PQ=4,\ PR=7,\ PM=\frac{7}{2}$

Solution:

Apollonius’s theorem gives $\begin{aligned} QR^2 &=2(4^2+7^2)-4\left(\frac72\right)^2\\ &=81. \end{aligned}$ Therefore $QR=9.$

Thus, the correct answer is D.

1964 AMC 12 Problem 38

Answer: D

Difficulty rating: 1210

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years