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The areas of two similar triangles are $169\ cm^2$ and $121\ cm^2$ respectively. If the longest side of the larger triangle is $26\ cm$, find the longest side of the smaller triangle.
Given:
The areas of two similar triangles are $169\ cm^2$ and $121\ cm^2$ respectively.
The longest side of the larger triangle is $26\ cm$.
To do:
We have to find the longest side of the smaller triangle.
Solution:
Let the longer side of the smaller triangle be $x$.
We know that,
The ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
$\frac{ar(larger\ triangle)}{ar(smaller\ triangle)} = (\frac{\ side\ of\ the\ larger\ triangle}{\ side\ of\ the\ smaller\ triangle})^2$
$= \frac{169}{121}$
$(\frac{\ side\ of\ the\ larger\ triangle}{\ side\ of\ the\ smaller\ triangle})=\sqrt{\frac{169}{121}}$
$= \frac{13}{11}$
Sides of similar triangles are proportional, this implies,
$\frac{(longer\ side\ of\ the\ larger\ triangle)}{(longer\ side\ of\ the\ smaller\ triangle)}=\frac{13}{11}$
$\frac{26}{x}=\frac{13}{11}$
$x = 2\times11$
$x=22\ cm$
The longest side of the smaller triangle is $22\ cm$.