Problem:
In β³ABC,M is the midpoint of side BC,AN bisects β BAC,BNβ₯AN and ΞΈ is the measure of β BAC. If sides AB and AC have lengths 14 and 19, respectively, then length MN equals
Answer Choices:
A. 2
B. 25β
C. 25ββsinΞΈ
D. 25ββ21βsinΞΈ
E. 25ββ21βsin(21βΞΈ)
Solution:
In the adjoining figure, BN is extended past N and meets AC at E. Triangle BNA is congruent to β³ENA, since β BAN=β EAN,AN=AN and β ANB=β ANE.
Therefore N is the midpoint of BE and AB=AE=14. Thus EC=5. Since M is given to be the midpoint of BC,MN joins the midpoints of two sides of β³BEC and MN =21β(EC)=25β.
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