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In triangle $ABC$ the three midpoints of the sides are $P, Q, R$. The midpoints of sides in triangle $PQR$ are $K, L, M$. What is the area of triangle $ABC$ if the area of triangle $KLM$ is $5$?

I started by drawing a picture with all the information. This gave me a a big triangle split into $4$ smaller triangles with the one in the middle being split again into $4$ pieces. If that one little piece has area $5$, do you get $ 5*2^2 $ as area for the medium triangle, so also do you get $5*4^2=80$ for the whole thing? Is there a way to prove that the areas of the split triangles are the same?

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Do you know that triangle formed by joining mid-points is similar to triangle given.

By similarity, its base and height are both half of the original triangle. So $area =1/2*base*height$ is $1/4$ original. (As both base and height are halved)

So area of $PQR=4*$Area of $KLM=20$

Area of $ABC=4*$area of $PQR=80$

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HInt: line joining mid points is parallel to 3rd side and half of 3rd side. can u prove that? rest will follow

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You can say that the two triangles $ABC$ and $KLM$ are similar by the intercept theorem with similarity ratio $4$, so the ratio between the two areas is $16$ and $ABC(area)=80$.

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