Problem 284

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Problem 284: (2018 AMC 12a, 23)

In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$ $\textbf{(A) } 76 \qquad$ $ \textbf{(B) } 77 \qquad$ $ \textbf{(C) } 78 \qquad$ $ \textbf{(D) } 79 \qquad$ $ \textbf{(E) } 80$

Solution

Exercise 284: Solution

In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=60^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$

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