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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 $JQR$?

Jun 25, 2024

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Since triangle PQR is the medial triangle of triangle MNO, QR is equidistant from NO and KL. Similarly, NO is equidistant from J and KL. Therefore, the altitude (J to QR) of triangle JQR is three times the altitude from P to QR. With the same base, the area of triangle JQR is three times triangle PQR which is 3 times 10, 30.

However, I have a question for you. Since triangle PQR is 10, and based on the information given in the question, I believe that MNO cannot be 20. If you draw out the diagram, it will be clearer that triangle MNO is four times triangle PQR.

Jun 25, 2024