ΔABC and ΔPQR are two similar triangles shown in the figure. AM and PN are the medians on ΔABC and ΔPQR, respectively. The ratio of areas of ΔABC and ΔPQR is 9:25. If AM = PO = 5 cm. Find the value of 3(ON).
A
15 cm
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B
10 cm
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C
21 cm
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D
18 cm
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Solution
The correct option is B 10 cm We are given two triangles ΔABC and ΔPQR such that ΔABC∼ΔPQR. ⇒ Corresponding sides will be proportional. ⇒ABPQ=BCQR=ACPR
Here, 12×BC12×QR=BMQN ∴ABPQ=BMQN
Since both the triangles ΔABC and ΔPQR are similar, there angles will be equal ⇒∠ A = ∠ P, ∠ B = ∠ Q and ∠ C = ∠ R
In ΔABM and ΔPQN, ABPQ=BMQN (Proved above) ∠ B = ∠ Q ∴ΔABM∼ΔPQN [By SAS similarity]
Given: AM = PO = 5 cm ⇒AreaofΔABCAreaofΔPQR=AB2PQ2=925 ⇒ABPQ=35 ⇒ABPQ=AMPN=35 ⇒55+ON=35 ⇒25=15+3(ON) ON=103 3(ON)=10cm
Hence the length of 3(ON) is 10 cm.