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 medians AD to E and PM to N such that AD = DE and PM = MN. Show triangles DAB and DEC are congruent (SAS), similarly MPQ and MNR. Using given proportionality, establish corresponding sides proportional and angles equal. By SSS similarity, the triangles are similar.

Step-by-step working

1
Draw correct figures and extend medians
1 mark
2
Prove $△ D A B ≅ △ D EC$ by SAS, similarly for PQR
1 mark
3
Use given proportionality to show sides of extended triangles proportional
1 mark
4
Prove $△ C A E \sim △ RPN$ by SSS
1 mark
5
Conclude $∠ B A C = ∠ QPR$ and by SAS, $△ A BC \sim △ PQR$
1 mark

Concepts used

similaritysss criterionmediantriangles

Approach

Step-by-step working

Concepts used

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