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)

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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