Question

It is given that sides AB and AC and median AD of $△ A BC$ are respectively proportional to sides PQ and PR and median PM of another $△ PQR$ . Show that $△ A BC \sim △ PQR$ .

Approach

Extend AD to E and PM to N such that AD = DE and PM = MN. Prove △DAB ≅ △DEC and △MPQ ≅ △MNR by SAS. Then AB = CE and PQ = NR. Given AB/PQ = AD/PM = AC/PR, substitute to get CE/NR = AE/PN = AC/PR. Hence △CAE ∼ △RPN by SSS. Therefore ∠BAC = ∠QPR, and △ABC ∼ △PQR by SAS similarity.

Step-by-step working

1
Draw correct figure with construction
1 mark
2
Prove △DAB ≅ △DEC and △MPQ ≅ △MNR, get AB = CE and PQ = NR
1 mark
3
Show CE/NR = AE/PN = AC/PR
0.5 marks
4
Prove △CAE ∼ △RPN by SSS, get ∠1 = ∠2 and ∠3 = ∠4
1 mark
5
∠BAC = ∠QPR, hence △ABC ∼ △PQR by SAS similarity
0.5 marks

Concepts used

similaritymediansproportionalitysss similarity

Approach

Step-by-step working

Concepts used

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