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) $△ BMC \sim △ QNR$
[Diagram shows two similar triangles ABC and PQR with medians CM and RN.]

Approach

Given $△ A BC \sim △ PQR$ , so $\frac{A B}{PQ} = \frac{BC}{QR} = \frac{A C}{PR}$ and corresponding angles are equal. Since CM and RN are medians, $A M = \frac{A B}{2}$ , $PN = \frac{PQ}{2}$ . (i) In $△ A MC$ and $△ PNR$ : $\frac{A M}{PN} = \frac{A C}{PR}$ and $∠ A = ∠ P$ , so $△ A MC \sim △ PNR$ by SAS similarity. (ii) From similarity, corresponding angles are equal, so $∠ BCM = ∠ QRN$ . (iii) Similarly, $△ BMC \sim △ QNR$ by SAS similarity.

Step-by-step working

1
Given $△ A BC \sim △ PQR$
1 mark
2
(i) In $△ A MC$ and $△ PNR$ : $\frac{A M}{PN} = \frac{A C}{PR}$ , $∠ A = ∠ P$
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3
(ii) From (i), $∠ A CM = ∠ PRN$ , so $∠ BCM = ∠ QRN$
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(iii) Similarly prove $△ BMC \sim △ QNR$
1.5 marks

Concepts used

similar trianglesmediansaa similaritycpct

Approach

Step-by-step working

Concepts used

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