# Prove that the ra

We have Given: ∆ABC ∆PQR

To prove: Ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

i.e., Construction: Draw AM perpendicular to BC and PN perpendicular to QR.

Proof: We know area of the triangle is given by (1/2 × base × height).

So in ∆ABC,

Area (∆ABC) = 1/2 × BC × AM [ base = BC & height = AM] …(i)

Similarly, in ∆PQR

Area (∆PQR) = 1/2 × QR × PN [ base = QR & height = PN] …(ii)

Dividing equations (i) by (ii), we get  …(iii)

In ∆ABM & ∆PQN,

B = Q [ ∆ABC ∆PQR; and corresponding angles are equal of similar triangles]

AMB = PNQ [ they are 90°]

So by AA-similarity property of triangle, ∆ABM ∆PQN. [ corresponding sides of similar triangles are proportional] …(iv)

Substituting equation (iv) in equation (iii), we have  …(v)

We know that,

∆ABC ∆PQR

Using property which says that corresponding sides of similar triangles are proportional, we can write as Using this equality, re-write equation (v)   …(A)

Using this equality again, re-writing equation (v)   …(B)

Similarly, …(C)

Collecting equations (A), (B) & (C), we get Hence, proved.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos  All Grammar41 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

<img width=RD Sharma - Mathematics

<img width=RD Sharma - Mathematics

If <img widRD Sharma - Mathematics

The areas of two RD Sharma - Mathematics

The areas of two RD Sharma - Mathematics

In Fig. 4.236, <sRD Sharma - Mathematics

If <img widRD Sharma - Mathematics

If the altitude oRD Sharma - Mathematics

If the areas of tRD Sharma - Mathematics