# Triangle A has sides of lengths 18 , 24 , and 12 . Triangle B is similar to triangle A and has a side of length 7 . What are the possible lengths of the other two sides of triangle B?

Oct 18, 2016

$\left(7 , \frac{28}{3} , \frac{14}{3}\right) , \left(\frac{21}{4} , 7 , \frac{7}{2}\right) , \left(\frac{21}{2} , 14 , 7\right)$

#### Explanation:

Anyone of the 3 sides of triangle B could be of length 7, hence there are 3 different possibilities for the sides of B.

Since the triangles are similar then the $\textcolor{b l u e}{\text{ratios of corresponding sides are equal}}$

Name the 3 sides of triangle B- a , b and c to correspond with the sides- 18 , 24 and 12 in triangle A.
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If side a = 7 then ratio of corresponding sides $= \frac{7}{18}$

and side b $= 24 \times \frac{7}{18} = \frac{28}{3} , \text{ side c } = 12 \times \frac{7}{18} = \frac{14}{3}$

The 3 sides of B would be $\left(7 , \textcolor{red}{\frac{28}{3}} , \textcolor{red}{\frac{14}{3}}\right)$
$\textcolor{b l u e}{\text{--------------------------------------------------------------}}$

If side b = 7 then ratio of corresponding sides $= \frac{7}{24}$

and side a $= 18 \times \frac{7}{24} = \frac{21}{4} , \text{ side c } = 12 \times \frac{7}{24} = \frac{7}{2}$

The 3 sides of B would be $\left(\textcolor{red}{\frac{21}{4}} , 7 , \textcolor{red}{\frac{7}{2}}\right)$
$\textcolor{b l u e}{\text{-------------------------------------------------------------------}}$

If side c = 7 then ratio of corresponding sides $= \frac{7}{12}$

and side a $= 18 \times \frac{7}{12} = \frac{21}{2} , \text{ side b } = 24 \times \frac{7}{12} = 14$

The 3 sides of B would be $\left(\textcolor{red}{\frac{21}{2}} , \textcolor{red}{14} , 7\right)$
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