Question

# Find the ratio of the areas of two similar triangles ΔABC and ΔPQR shown in the figure.

A
(ABPQ)2
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B
(BCQR)2
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C
(ACPR)2
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D
All of these
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Solution

## The correct option is D All of theseWe are given two triangles ABC and PQR such that ΔABC∼ΔPQR. For finding areas of the two triangles, we draw altitudes AM and PN of the triangles. Ar(ΔABC)=12×BC×AM and Ar(ΔPQR)=12×QR×PN ⇒ Ar(ΔABC)Ar(ΔPQR)=12×BC×AM12×QR×PN=BC×AMQR×PN Now, in ΔABM and ΔPQN, ∠B=∠Q (As ΔABC∼ΔPQR) and ∠M=∠N (Each is of 90∘) So, ΔABM∼ΔPQN (AA similarity) ∴ AMPN=ABPQ Also, ΔABC∼ΔPQR So, ABPQ=BCQR=CAPR ∴ Ar(ΔABC)Ar(ΔPQR) = ABPQ×AMPN = ABPQ×ABPQ = (ABPQ)2 Since the ratio of sides is the same, Ar(ΔABC)Ar(ΔPQR) = (ABPQ)2 = (BCQR)2 = (ACPR)2 If ΔABC and ΔPQR are similar, then the ratio of their areas will be equal to the square of the ratio of their corresponding sides.

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