Spreading the side equally

Geometry Level 3

Each side of triangle $ABC$ is extended such that $AB=BP, BC=CQ, CA=AR,$ as shown in the diagram above.

If the area of triangle $ABC$ is 10, then what is the area of triangle $PQR?$

The answer is 70.

1 solution

Nikkil V
Dec 6, 2016

In $\Delta(ACP)$ , $CB$ is the median.

So,Area of $\Delta(ABC)$ = Area of $\Delta(PBC)$

Now in $\Delta(PQB)$ , $PC$ is the median,

So, Area of $\Delta(PQC)$ = Area of $\Delta(PBC)$

This implies, Area of $\Delta(PQC)$ = Area of $\Delta(ABC)$

So, Area of $\Delta(PBQ)$ = 2 x Area of $\Delta(ABC)$

Similarly, Area of $\Delta(RAP)$ = Area of $\Delta(RCQ)$ =2 x Area of $\Delta(ABC)$

Therefore,Area of $\Delta(PQR)$ = Area of $\Delta(ABC)$ + Area of $\Delta(PQB)$ +Area of $\Delta(RAP)$ + Area of $\Delta(RCQ)$

Area of $\Delta(PQR)$ = $7$ x Area of $\Delta(ABC)$

Area of $\Delta(PQR)$ = 7 x 10 units = 70 units

Thanks for cleaning up the solution. This makes it much easier to read and understand.

Calvin Lin Staff - 4 years, 6 months ago

Nice solution dude ..can't we do it by assuming it right angled trialngle ..???? Pls reply

Satyam Tripathi - 4 years, 6 months ago

Yes, that would technically give you the correct numerical answer.

However, it does not constitute a rigorous proof of the problem. At the end of the day, understanding the thought process / approach of solving the problem is much more important than what the actual answer is.

Calvin Lin Staff - 4 years, 6 months ago

Thanks @Satyam Tripathi

Nikkil V - 4 years, 6 months ago

when I first saw this problem I thought it quite hard .But when ever I solved it, I understood, this problem was not so tough but tricky ..

Abdullah Ahmed - 4 years, 6 months ago

U are welcome Nikhil ..BTW are u studying in fiitjee

Satyam Tripathi - 4 years, 6 months ago

No I am not studying in FIITJEE

Nikkil V - 4 years, 6 months ago

Spreading the side equally

The answer is 70.

1 solution

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