In △ ABC, M is the midpoint of side BC, AN bisects ∠ BAC, BN⊥ AN, and θ is the measure of ∠ BAC. If sides AB and AC have lengths 14 and 19, respectively, then length MN equals

1981 AMC 12 Problem 19

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19.

\triangle ABC,

M

is the midpoint of side

BC,

AN

bisects

\angle BAC,

BN\perp AN,

and

\theta

is the measure of

\angle BAC.

If sides

AB

and

AC

have lengths

14

and

19,

respectively, then length

MN

equals

2

\frac52

\frac52-\sin\theta

\frac52-\frac12\sin\theta

\frac52-\frac12\sin\left(\frac\theta2\right)

Extend

BN

through

N

to meet

AC

Use congruence across the angle bisector, then apply the midpoint theorem

Extend

BN

to meet

AC

E.

The right triangles

ABN

and

AEN

are congruent because

AN

is common and bisects the angle at

A.

Hence

BE

is bisected by

N

and

AE=AB=14.

Since

AC=19,

EC=5.

In triangle

BEC,

points

N

and

M

are the midpoints of

BE

and

BC,

MN=\frac{EC}{2}=\frac{5}{2}.

Therefore, the correct answer is B.