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

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

Class 10 | Exercise 15E | ICSE | Maths | Selina | Similarity (With Applications to Maps and Models)

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

Selina Solutions Concise Class 10 Maths Chapter 15 ex. 15(E) - 13

Solution:

In

$\vartriangle ABC,\text{ }as\text{ }PR\text{ }||\text{ }BC\text{ }by\text{ }BPT$

we have

$AP/PB\text{ }=\text{ }AR/RC$

And, in

$\vartriangle PAR\text{ }and\text{ }\vartriangle BAC,$

$\angle PAR\text{ }=\angle BAC$

[Common]

$\angle APR\text{ }=\angle ABC$

[Corresponding angles]

Hence,

$\vartriangle PAR\text{ }\sim\text{ }\vartriangle BAC\text{ }by\text{ }AA$

criterion for similarity

So, we have

$PR/BC\text{ }=\text{ }AP/AB$

$PR/BC\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$

[Since, P is the mid-point of AB]

$PR\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }BC$

Similarly,

$PQ\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }AC$

$RQ\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }AB$

So,

$PR/BC\text{ }=\text{ }PQ/AC$

$=\text{ }RQ/AB$

Therefore,

$\vartriangle QRP\text{ }\sim\text{ }\vartriangle ABC\text{ }by\text{ }SSS$

similarity.

i

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