+12 Points M, N, and O, are the midpoints of sides KL, LJ, and JK, respectively, of triangle JKL. Points P, Q, and R are the midpoints of NO, OM, and MN, respectively. If the area of triangle PQR is 12, then what is the area of triangle JQR?
+310 We have a triangle JKL with midpoints M, N, and O of its sides. Then, we have another triangle PQR formed by the midpoints of the sides of triangle MNO.
The area of triangle PQR is given as 12. We need to find the area of triangle JQR.
Key Concept
Midpoint Theorem: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half its length.
Area Scaling: If the sides of a triangle are scaled by a factor of 'k', then its area is scaled by a factor of 'k^2'.
Solution
Triangle MNO and Triangle JKL:
Since M, N, and O are midpoints of JK, KL, and LJ respectively, triangle MNO is similar to triangle JKL with a scale factor of 1/2 (due to the midpoint theorem).
Therefore, the area of triangle MNO is (1/2)^2 = 1/4 times the area of triangle JKL.
Triangle PQR and Triangle MNO:
Similarly, triangle PQR is similar to triangle MNO with a scale factor of 1/2.
So, the area of triangle MNO is (1/2)^2 = 1/4 times the area of triangle PQR.
Finding the area of triangle JKL:
We know the area of triangle PQR is 12.
So, the area of triangle MNO is 12 * 4 = 48.
Therefore, the area of triangle JKL is 48 * 4 = 192.
Finding the area of triangle JQR:
Triangle JQR is half of triangle JKL (since QR is a median).
So, the area of triangle JQR is 192 / 2 = 96.
Therefore, the area of triangle JQR is 96.
