Question

The areas of two similar triangles ABC and DEF are 144 square cm and 81 square cm respectively. If the longest side of the larger triangle ABC is 36 cm, then the longest side of the smallest triangle DEF is
A. 20 cm
B. 26 cm
C. 27 cm
D. 30 cm

Answer 1

Hint: The theorem to be used is an area of similar triangles theorem. According to the theorem, if \[\Delta ABC\] and \[\Delta DEF\] are similar to each other, then ratio of the area of the triangle is equal to square of the ratios of their corresponding sides as,
\[\dfrac{{Area(\Delta ABC)}}{{Area(\Delta DEF)}} = {\left( {\dfrac{{AB}}{{DE}}} \right)^2} = {\left( {\dfrac{{BC}}{{EF}}} \right)^2} = {\left( {\dfrac{{CA}}{{FD}}} \right)^2}\]

Complete step by step solution:
Given,

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.