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# help geometry

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Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$.  Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively.  If the area of triangle $PQR$ is $10$, and the area of triangle $MNO$ is $20$, then what is the area of triangle $JQR$?

Apr 15, 2024

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Since P, Q, and R are the midpoints of segments in triangle MNO, then triangle PQR is similar to triangle MNO with a scale factor of 21​. Therefore, the area of triangle PQR is (21​)2 times the area of triangle MNO, or 41​ the area of triangle MNO. We are given that the area of triangle PQR is 10, so the area of triangle MNO is 10⋅4=40.

Similarly, triangle JQR is similar to triangle MNO with a scale factor of 21​, so the area of triangle JQR is (21​)2 times the area of triangle MNO, or 41​ the area of triangle MNO. We know the area of MNO is 40, so the area of triangle JQR is 41​⋅40=10​.

Apr 15, 2024