In the adjoining figure triangle ABC is such that AB=4 and AC=8. If M is the midpoint of BC and AM=3, what is the length of BC?

1975 AMC 12 Problem 20

Attempt Problem 20 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

20.

In the adjoining figure triangle

ABC

is such that

AB=4

and

AC=8.

M

is the midpoint of

BC

and

AM=3,

what is the length of

BC?

2\sqrt{26}

2\sqrt{31}

9

4+2\sqrt{13}

not enough information given to solve the problem

Use the relation between a triangle’s three sides and a median

Apollonius’ theorem gives

AB^2+AC^2=2(AM^2+BM^2)

Let

BM=CM=x,

BC=2x.

By Apollonius’ theorem,

4^2+8^2=2(3^2+x^2).

Hence

80=18+2x^2,\ x^2=31,

and

BC=2\sqrt{31}.

Therefore, the correct answer is B.