Triangle A has sides of lengths 51 , 48 , and 54 . Triangle B is similar to triangle A and has a side of length 3 . What are the possible lengths of the other two sides of triangle B?

May 9, 2016

$\left(3 , \frac{48}{17} , \frac{54}{17}\right) , \left(\frac{51}{16} , 3 , \frac{27}{8}\right) , \left(\frac{17}{6} , \frac{8}{3} , 3\right)$

Explanation:

Since triangle B has 3 sides , anyone of them could be of length 3 and so there are 3 different possibilities.
Since the triangles are similar then the ratios of corresponding sides are equal.
Name the 3 sides of triangle B, a ,b and c , corresponding to the sides 51 ,48 ,54 in triangle A.
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If side a = 3 then ratio of corresponding sides $= \frac{3}{51} = \frac{1}{17}$
hence b$= 48 \times \frac{1}{17} = \frac{48}{17} \text{ and } c = 54 \times \frac{1}{17} = \frac{54}{17}$
The 3 sides of B $= \left(3 , \frac{48}{17} , \frac{54}{17}\right)$
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If side b = 3 then ratio of corresponding sides $= \frac{3}{48} = \frac{1}{16}$
hence a$= 51 \times \frac{1}{16} = \frac{51}{16} \text{ and } c = 54 \times \frac{1}{16} = \frac{27}{8}$
The 3 sides of B $= \left(\frac{51}{16} , 3 , \frac{27}{8}\right)$
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If side c = 3 then ratio of corresponding sides $= \frac{3}{54} = \frac{1}{18}$
hence a $= 51 \times \frac{1}{18} = \frac{17}{6} \text{ and } b = 48 \times \frac{1}{18} = \frac{8}{3}$
The 3 sides of B $= \left(\frac{17}{6} , \frac{8}{3} , 3\right)$
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