Question

In the given triangle P, Q and R are the mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

Answer 1

In ∆ABC, PR || BC.

By Basic proportionality theorem,

`(AP)/(PB) = (AR)/(RC)`

Also, in ∆PAR and ∆ABC,

∠PAR = ∠BAC   ...(Common)

∠APR = ∠ABC   ...(Corresponding angles)

∆PAR ~ ∆BAC   ...(AA similarity)

`(PR)/(BC) = (AP)/(AB)`

`(PR)/(BC) = 1/2`   ...(As P is the mid-point of AB)

`(PR)/(BC) = 1/2 BC`

Similarity, `PQ = 1/2 AC`

`RQ = 1/2 AB`

Thus, `(PR)/(BC) = (PQ)/(AC) = (RQ)/(AB)`

`=>` ∆QRP ~ ∆ABC  ...(SSS similarity)