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Question

# In ΔABC, point P,Q and R are the mid points of the sides AB, BC and CA respectively. If area of ΔABC is 32 sq units, then area of ΔPQR is

A
8 sq cm
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B
16 sq cm
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C
64 sq cm
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D
24 sq cm
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Solution

## The correct option is A 8 sq cmThe line joining the midpoints of the sides of the triangle form four triangles, each of which is similar to the original triangle.ΔABC∼ΔPQRIn ΔABC, P and R are mid points of AB and AC respectively.∴PR||BC (midpoint theorem)In ΔABC and ΔAPR:∠A is common and ∠APR=∠ABC (corresponding angles)Therefore, ΔABC∼ΔAPR (AA similarity)In ΔABC and ΔPQR, since P,Q,R are the midpoints of AB,BC and AC respectively,PR=12BC; (midpoint theorem)∴ΔABC∼ΔPQR (SSS similarity)Ar(ΔPQR)Ar(ΔABC)=PR2BC2=(12)2=14 Now, it is given that area of ΔABC is 32 sq. units. Therefore, we have:Ar(ΔPQR)32=14Ar(ΔPQR)=324Ar(ΔPQR)=8Hence, the area of ΔPQR is 8 sq units.

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