1964 AMC 12 Problem 38

Attempt Problem 38 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

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38.

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.

Problem 38 in other years