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

Since ΔABC ~ ΔPQR, corresponding sides are proportional and angles equal. Medians divide sides in 2:1 ratio. Using these facts and AA similarity, all three parts can be proven by showing angle and side ratio conditions.

Step-by-step working

1
ΔABC ~ ΔPQR implies AB/PQ = AC/PR and ∠A = ∠P
1 mark
2
Since M, N are midpoints, AM/PN = AB/(2PQ)
1.5 marks
3
ΔAMC ~ ΔPNR by SAS similarity
1 mark
4
∠BCM = ∠QRN from similarity, ΔBMC ~ ΔQNR
1.5 marks

Concepts used

similar trianglesmediansproof

Approach

Step-by-step working

Concepts used

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