Triangle A has sides of lengths $12$ $12$ , $16$ $16$ , and $18$ $18$ . Triangle B is similar to triangle A and has a side with a length of $16$ $16$ . What are the possible lengths of the other two sides of triangle B?

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Triangle A has sides of lengths $12$ $12$ , $16$ $16$ , and $18$ $18$ . Triangle B is similar to triangle A and has a side with a length of $16$ $16$ . What are the possible lengths of the other two sides of triangle B?

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Answer 1 · 2017-01-23T21:31:58

There are 3 possible sets of lengths for Triangle B.

Explanation:

For triangles to be similar, all sides of Triangle A are in the same proportions to the corresponding sides in Triangle B.

If we call the lengths of the sides of each triangle { $A_{1}$ $A_1$ , $A_{2}$ $A_2$ , and $A_{3}$ $A_3$ } and { $B_{1}$ $B_1$ , $B_{2}$ $B_2$ , and $B_{3}$ $B_3$ }, we can say:

$\frac{A_{1}}{B_{1}} = \frac{A_{2}}{B_{2}} = \frac{A_{3}}{B_{3}}$ $A_1/B_1 = A_2/B_2 = A_3/B_3$

or

$\frac{12}{B_{1}} = \frac{16}{B_{2}} = \frac{18}{B_{3}}$ $12/B_1 = 16/B_2 = 18/B_3$

The given information says that one of the sides of Triangle B is 16 but we don't know which side. It could be the shortest side ( $B_{1}$ $B_1$ ), the longest side ( $B_{3}$ $B_3$ ), or the "middle" side ( $B_{2}$ $B_2$ ) so we must consider all possibilities

If $B_{1} = 16$ $B_1=16$

$\frac{12}{16} = \frac{3}{4}$ $12/color(red)(16) = 3/4$
$\frac{3}{4} = \frac{16}{B_{2}} \Rightarrow B_{2} = 21.333$ $3/4 = 16/B_2 => B_2=21.333$
$\frac{3}{4} = \frac{18}{B_{3}} \Rightarrow B_{3} = 24$ $3/4 = 18/B_3 => B_3=24$

{16, 21.333, 24} is one possibility for Triangle B

If $B_{2} = 16$ $B_2=16$

$\frac{16}{16} = 1 \Rightarrow$ $16/color(red)(16) = 1 =>$ This is a special case where Triangle B is exactly the same as Triangle A. The triangles are congruent.

{12, 16, 18} is one possibility for Triangle B.

If $B_{3} = 16$ $B_3=16$

$\frac{18}{16} = \frac{9}{8}$ $18/color(red)(16) = 9/8$
$\frac{9}{8} = \frac{12}{B_{1}} \Rightarrow B_{1} = 10.667$ $9/8 = 12/B_1 => B_1 = 10.667$
$\frac{9}{8} = \frac{16}{B_{2}} \Rightarrow B_{2} = 14.222$ $9/8 = 16/B_2 => B_2 = 14.222$

{10.667, 14.222, 16} is one possibility for Triangle B.

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Triangle A has sides of lengths $12$ $12$ , $16$ $16$ , and $18$ $18$ . Triangle B is similar to triangle A and has a side with a length of $16$ $16$ . What are the possible lengths of the other two sides of triangle B?

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Triangle A has sides of lengths 1212 , 1616 , and 1818 . Triangle B is similar to triangle A and has a side with a length of 1616 . What are the possible lengths of the other two sides of triangle B?

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Explanation:

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Triangle A has sides of lengths $12$ $12$ , $16$ $16$ , and $18$ $18$ . Triangle B is similar to triangle A and has a side with a length of $16$ $16$ . What are the possible lengths of the other two sides of triangle B?