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 $JQR$?

Akhain1 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.

Boseo Apr 15, 2024