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# Quick Question

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Hey y'all, it's me again.

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 27, then what is the area of triangle LPQ?

Any assistance is greatly appreciated!

Oct 23, 2022

#1
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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 27, then what is the area of triangle LPQ?

Note:

The line joining the midpoints of a two sides of a triangle is parallel to the 3rd side. You can look up the proof if you wish.

So

KJ is parallel to MN is parallel to QP

ans

KL is parallel to ON is parallel to QR

So every triangle in this figure is similar to all the others.

LPQ=27u^2

So the area of MNO = 4*27 =  108u^2

And the area of LJK = 4*108 = 432u^2

Before I think on it any further can you confirm that it really is triangle LPQ that you want the area of ...

Here is the diagram.

Oct 23, 2022
#2
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Yes, LPQ is the triangle we want the area of. I might be able to post a picture of the question, if you'd like.

Qube73  Oct 23, 2022
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ok

triangle LMN is congruent to triangle ONM  so LM=NO

And NP is a half of NO

And RQ=NP

So the horizontal distance from Q to L is 3 times the horizontal distance from Q to R

Triangles  RPQ and LPQ share the same base, that is PQ

But the perpendicular height of LPQ is three times that of RPQ

Since the area of triangle RPQ is  27,

It follows that the area of LPQ is  3*27=  81u^2

Oct 24, 2022
#4
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Thank so much!

Qube73  Oct 24, 2022
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You are welcome.

Melody  Oct 24, 2022