The areas of two similar triangles abc and def are 36 and 81

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Given: Areas of two similar triangles ΔABC and ΔDEF are 144cm2 and 81cm2.

If the longest side of larger ΔABC is 36cm

To find: the longest side of the smaller triangle ΔDEF

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

`\text{ar(Δ ABC)}/\text{ar(Δ DEF)}=(\text{longest side of larger Δ ABC}/\text{longest side of smaller Δ DEF})^2`

`114/81=(36/\text{longest side of smaller Δ DEF})^2`

Taking square root on both sides, we get

\[\frac{12}{9} = \frac{36}{\text{longest side of smaller ∆ DEF}}\]

`\text{longest side of smaller Δ DEF}=(36xx9)/12=27cm`

Hence the correct answer is `C`

Answer: (3) 27 cm

If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

Ratio of areas = 144:81

Ratio of sides = 12:1

Let the longest side be X

Then

36 : X = 12:9

X = (36 × 9)/12

X = 27 cm

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We have, Area of \[\Delta ABC = 144c{m^2}\] Area of \[\Delta DEF = 81c{m^2}\]Also \[AB = 36cm\].The theorem which relates the area of two similar triangles in terms of its side is area of similar triangle theorem. It states that for two similar triangles the ratio of their areas is equal to ratio of the square of their corresponding sides.\[\dfrac{{Area(\Delta ABC)}}{{Area(\Delta DEF)}} = {\left( {\dfrac{{AB}}{{DE}}} \right)^2} = {\left( {\dfrac{{BC}}{{EF}}} \right)^2} = {\left( {\dfrac{{CA}}{{FD}}} \right)^2}\]It can also be written as,\[ \Rightarrow \dfrac{{Area(\Delta ABC)}}{{Area(\Delta DEF)}} = {\left( {\dfrac{{AB}}{{DE}}} \right)^2}\]Substituting the given values we have,\[ \Rightarrow \dfrac{{144}}{{81}} = {\left( {\dfrac{{36}}{{DE}}} \right)^2}\]Rearranging we have,\[ \Rightarrow {\left( {\dfrac{{36}}{{DE}}} \right)^2} = \dfrac{{144}}{{81}}\]Taking square root on both side we have,\[ \Rightarrow \left( {\dfrac{{36}}{{DE}}} \right) = \sqrt {\dfrac{{144}}{{81}}} \]\[ \Rightarrow \dfrac{{36}}{{DE}} = \dfrac{{12}}{9}\]Taking reciprocal of whole equation,\[ \Rightarrow \dfrac{{DE}}{{36}} = \dfrac{9}{{12}}\]\[ \Rightarrow DE = \dfrac{9}{{12}} \times 36\]\[ \Rightarrow DE = 9 \times 3\]\[ \Rightarrow DE = 27cm\]Hence the length of the longest side of a triangle \[\Delta DEF\] is 27 cm.

Hence the required answer is option (c).

Note: Triangles are similar if they have the same shape, but can be different sizes. The important step in the question is the use of areas of similar triangle theorems.

The important properties of the triangle are,Corresponding angles are congruent (same measure).Corresponding sides are all in the same proportion.

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