Question

In the given figure, CM and RN are respectively, the medians of $△ A BC$ and $△ PQR$ . If $△ A BC \sim △ PQR$ , prove that:
(i) $△ A MC \sim △ PNR$
(ii) $∠ BCM = ∠ QRN$
(iii) $△ BMC \sim △ QNR$

Approach

Using similarity of ABC and PQR, we show corresponding sides are proportional. For medians, AC/PR = (1/2)AB/(1/2)PQ = AM/PN. With angle A = angle P, triangle AMC ~ PNR by SAS. From cpct, angle ACM = angle PRN. Since angle ACB = angle PRQ, angle BCM = angle QRN. Similarly, triangle BMC ~ QNR.

Step-by-step working

1
$△ A BC \sim △ PQR \Rightarrow \frac{A B}{PQ} = \frac{A C}{PR}$ . Hence $\frac{A C}{PR} = \frac{A M}{PN}$ and $∠ A = ∠ P$
1 mark
2
$△ A MC \sim △ PNR$ by SAS similarity criterion
1 mark
3
$△ A MC \sim △ PNR \Rightarrow ∠ A CM = ∠ PRN$ . Since $∠ A CB = ∠ PRQ$ , we get $∠ BCM = ∠ QRN$
1.5 marks
4
$\frac{BC}{QR} = \frac{BM}{QN}$ and $∠ B = ∠ Q \Rightarrow △ BMC \sim △ QNR$ by SAS
1.5 marks

Concepts used

similaritymedianstriangles

Approach

Step-by-step working

Concepts used

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