Q. 1

In ΔABC, such that are positive real numbers. A line passing through P and parallel to intersects in Q. Prove that (m + n)2 (area of ΔAPB) = m2(area of ΔABC)

Answer :

We have


Given: In ∆ABC, such that , where m and n are positive real numbers.


QP BC


Prove that: (m + n)2 (area of ∆APB) = m2 (area of ∆ABC)


Proof: Since


[by invertendo componendo invertendo rule]



In ∆APQ and ∆ABC,


APQ ABC [, corresponding angles]


AQP ACB [, corresponding angles]


The correspondence APQ ABC is a similarity.


Remember the property, areas of similar triangles are proportional to the squares of the corresponding sides.






By cross-multiplication, we get


(m + n)2 (Area of ∆APB) = m2 (Area of ∆ABC)


Thus, proved.


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