Triangle and Bisector

Geometry Level 3

In $\triangle ABC$ above, $M$ is the midpoint of side $BC,$ line segment $AN$ bisects $\angle BAC ,$ and $\overline{ AN } \bot \overline{ BN }.$ If $\overline{AB} =14$ and $\overline{AC} =19,$ what is $\overline{MN}?$

\frac{5}{4}

\frac{5}{2}

5

\frac{5}{3}

2 solutions

Max Sánchez
Jun 7, 2014

Extend BN until it crosses AC in B'. Like AN bisects angle BAC, BN=NB', then AB=AB'=14 Like BN=NB' and BM=MC, then triangle BNM its similar to triangle BB'C. Then, BB'=2(BN), so B'C=2(NM), 5=2(NM), NM=5/2

Rab Gani
Apr 20, 2014

Extend line BN so that it crosses AC at O. AO = 14, and CO = 5. Because BN=NO, then MN=5/2

Triangle and Bisector

2 solutions

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