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CAT Areas of Similar Triangles - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Quick Facts

  • 2 Questions around this concept.

Solve by difficulty

QuestionDIFFICULT

If triangles ABC and DBC are on the same base BC and AD intersects BC at O

then \frac{\operatorname{ar}(\Delta A B C)}{\operatorname{ar}(\Delta D B C)}

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In the given figure, the line segment XY is parallel to the side AC of the triangle ABC and it divides the triangle into two parts of equal area. Then AX:AB is

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Concepts Covered - 1

Areas of Similar Triangles

You have learnt that in two similar triangles, the ratio of their corresponding sides is the same. Do you think there is any relationship between the ratio of their areas and the ratio of the corresponding sides?

You know that area is measured in square units. So, you may expect that this ratio is the square of the ratio of their corresponding sides. This is indeed true and we shall prove it in the next theorem.

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

Given: \triangle A B C \sim \triangle D E F

To prove: \frac{\operatorname{ar}(\Delta A B C)}{\operatorname{ar}(\Delta D E F)}=\frac{A B^{2}}{D E^{2}}=\frac{A C^{2}}{D F^{2}}=\frac{B C^{2}}{E F^{2}}

Construction: \text{Draw } A L \perp B C\text{ and }D M \perp E F

Prrof: \triangle A B C \sim \triangle D E F

\therefore \quad \angle A=\angle D,\;\; \angle B=\angle E,\;\; \angle C=\angle F

\text { and } \frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}\qquad\ldots(i)

\text { Now, in }\Delta ABL\text{ and }\Delta DEM

\angle B =\angle E \qquad \text{(As } \Delta ABC \sim \Delta DEF )

\\ \angle L =\angle M \qquad \text { (Each is of }\left.90^{\circ}\right)

\therefore\quad\Delta ABL \sim \Delta DEM \qquad \text{(AA similarity criterion)}

\therefore\quad\frac{ AL }{ DM }=\frac{ AB }{ DE }\qquad\ldots(ii)

Form (i) and (ii)

\frac{AL}{DM}=\frac{B C}{E F}\qquad\ldots(iii)

\\\text { Now, } \operatorname{ar}(\triangle A B C)=\frac{1}{2} \times B C \times A L \\

\text { and }\;\; \operatorname{ar}(\triangle D E F)=\frac{1}{2} \times E F \times D M

\begin{aligned} \frac{\operatorname{ar}( ABC )}{\operatorname{ar}( DEF )} &=\frac{ AB }{ DE } \times \frac{ AL }{ DM }\\ &=\frac{ AB }{ DE } \times \frac{ AB }{ DE } \qquad\qquad\text{[From (iii)]} \\ &=\left(\frac{ AB }{ DE }\right)^{2} \end{aligned}

\text { Similarly, } \frac{\operatorname{ar}(\Delta A B C)}{\operatorname{ar}(\Delta D E F)}=\frac{A B^{2}}{D E^{2}} \text { and } \frac{\operatorname{ar}(\Delta A B C)}{\operatorname{ar}(\Delta D E F)}=\frac{A C^{2}}{D F^{2}} \\

\text { Hence, } \frac{\operatorname{ar}(\triangle A B C)}{\operatorname{ar}(\Delta D E F)}=\frac{A B^{2}}{D E^{2}}=\frac{A C^{2}}{D F^{2}}=\frac{B C^{2}}{E F^{2}}

Important Results on Area of Similar Triangle:

1. The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes. 

  

  In the following figures, ΔABC ~ ΔDEF and AX, DY are the altitudes.

  \text { Then, } \frac{\text { Area of } \Delta A B C}{\text { Area of } \Delta D E F}=\frac{A X^{2}}{D Y^{2}}

 

2. The ratio of areas of two similar triangles is equal to the ratio of the squares on their corresponding medians.

  

  In the following figure, ΔABC ~ ΔPQR and AD and PS are medians.

  \text { Then, } \frac{\text { Area of } \Delta A B C}{\text { Area of } \Delta P Q R}=\frac{A D^{2}}{P S^{2}}

 

3. The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisector segments.

  

  In the figure, ΔABC ~ ΔDEF and AP, DQ are bisectors of ∠A and ∠D respectively, 

  \text { Then, } \frac{\text { Area of } \Delta A B C}{\text { Area of } \Delta D E F}=\frac{A P^{2}}{D Q^{2}}

Study other Related Concepts

CAT Areas of Similar Triangles Current Topic

CAT Basic Terms and Definitions of Lines and Angles
CAT Various Types of Angle
CAT Linear Pair of Angles
CAT Vertically Opposite Angles
CAT Corresponding Angle Axioms
CAT Alternate Interior Angle Theorem
CAT Interior Angle Theorem
CAT Parallel and a Transversal Lines
CAT Introduction to Triangles
CAT Congruence of Triangles

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