1961 AMC 12 Problem 36

Question

In triangle

ABC

the median from

A

is perpendicular to the median from

B.

If

BC=7

and

AC=6,

find the length of

AB.

4

\sqrt{17}

4.25

2\sqrt5

4.5

Put

A

at the origin and represent

B

and

C

by vectors

Write direction vectors for the two medians and set their dot product to zero

Let

A=\mathbf0,

B=\mathbf b,

and

C=\mathbf c,

with

|\mathbf b|=\ell,

|\mathbf c|=6,

and

|\mathbf c-\mathbf b|=7.

The median directions are

\frac{\mathbf b+\mathbf c}{2}

from

A

and

\frac{\mathbf c}{2}-\mathbf b

from

B.

Their dot product is zero, so

(\mathbf b+\mathbf c)\mathbin{\cdot}(\mathbf c-2\mathbf b)=0.

Using

\mathbf b\mathbin{\cdot}\mathbf c=\frac{\ell^2+36-49}{2},

this simplifies to

\ell^2=17.

Hence

AB=\sqrt{17}.

Therefore, the correct answer is B.

Problem 36 in other years