Area of Similar Triangles


 
 
Concept Explanation
 

Area of Similar Triangles

Theorem: Ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

Given:  large Delta ABC sim Delta PQR

To Prove:

large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}

Construction: Draw AD large perp BC and PS large perp QR

Proof: In large Delta ABD and large Delta PQS

large angle B = angle Q                      large [;Delta ABC sim Delta PQR;]

large angle D = angle S                       large [ ;Each ;90^0;]

large Rightarrow Delta ABD sim Delta PQS                     [ By AA Similarity Criterion ]

Ratio of corresponding sides of similar triangle

large frac{AB}{PQ}=frac{AD}{PS}                     ..........................(1)

But we are given that large Delta ABC sim Delta PQR

    large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}                 ..........................(2)

From Eq (1) and (2)

large frac{BC}{QR}=frac{AD}{PS}

Now

large frac{area(Delta ABC)}{area(Delta PQR)}=frac{frac{1}{2}X BC;X;AD}{frac{1}{2}X QR;X;PS}            large [;area ;of ;Delta = frac{1}{2};X;base;X;height;]

large frac{area(Delta ABC)}{area(Delta PQR)}=frac{BC;X;AD}{QR;X;PS}=frac{BC^2}{QR^2}

As large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}

Hence  large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}

Illustration: If the area of two similar triangles are equal, prove that they are congruent.

Let large Delta ABC sim Delta PQR

large Rightarrow large frac{AB}{PQ}=frac{AC}{PR}=frac{BC}{QR}

Given      

large frac{area(Delta ABC)}{area(Delta PQR)}=1

We know that

large frac{area(Delta ABC)}{area(Delta PQR)}=frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}

large Rightarrow frac{AB^2}{PQ^2}=frac{BC^2}{QR^2}=frac{AC^2}{PR^2}=1

large Rightarrow AB = PQ , BC= QR and AC = PR

Hence

large Delta ABC cong Delta PQR     [By SSS Criterion of Congruence]

Sample Questions
(More Questions for each concept available in Login)
Question : 1

The ratio of any two corresponding sides of two similar triangles is 3:2; the ratio of the areas of the two triangles is _________________.

Right Option : C
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Explanation
Question : 2

The area of two isosceles triangles are in the ratio 16 : 25. The ratio of their corresponding heights is _______________

Right Option : B
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Explanation
Question : 3

In the following figure, bigtriangleup ABCsim bigtriangleup DEF. The area of bigtriangleup ABC is 9;cm^{2} and that of bigtriangleup DEF is  16;cm^{2}, EF = 4.2 cm. Find BC.

Right Option : C
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Explanation
 
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