Triangle A has sides of lengths #51 #, #45 #, and #54 #. Triangle B is similar to triangle A and has a side of length #9 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #51    #, #45    #, and #54     #. Triangle B is similar to triangle A and has a side of length #9     #. What are the possible lengths of the other two sides of triangle B?

Evelyn Arboleda · Accepted Answer

9, 8.5 & 7.5
9, 10.2 & 10.8
7.941, 9 & 9.529

If 9 is the longest side then the multiplier wold be #54/9=6# #51/6=8.5#. #45/6=7.5# If 9 is the shortest side then the multiplier would be #45/9=5# #51/5=10.2#, # 54/5=10.8# If 9 is the middle side then the multiplier would be #51/9=5 2/3# #45/(5 2/3)=7.941#, #54/(5 2/3)=9.529#

Maverick Smith · Answer

undefined The possible lengths of the other two sides of triangle B can be found by setting up a proportion based on the similarity of the two triangles.

The sides of triangle A are in the ratio $51 : 45 : 54$, which simplifies to $17 : 15 : 18$. Since triangle B is similar to triangle A, the corresponding sides are also in the ratio of $17 : 15 : 18$.

If the side length given for triangle B is 9, then the ratios of the sides of triangle B are $9 : x : y$, where $x$ and $y$ are the lengths of the other two sides.

Setting up the proportion:

$\frac{9}{x} = \frac{17}{15}$

Solving for $x$:

$17x = 9 \times 15$

$17x = 135$

$x = \frac{135}{17}$

Similarly, setting up another proportion for the third side:

$\frac{9}{y} = \frac{17}{18}$

Solving for $y$:

$17y = 9 \times 18$

$17y = 162$

$y = \frac{162}{17}$

Therefore, the possible lengths of the other two sides of triangle B are approximately $7.94$ and $9.53$.