# Triangle A has sides of lengths 36 , 42 , and 60 . 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?

Jun 20, 2016

$\left\{\textcolor{w h i t e}{\frac{2}{2}} \textcolor{m a \ge n t a}{7} \text{ ; "color(blue)(8.16bar6-> 8 1/6)" ; } \textcolor{b r o w n}{11.6 \overline{6} \to 11 \frac{2}{3}} \textcolor{w h i t e}{\frac{2}{2}}\right\}$

$\left\{\textcolor{w h i t e}{\frac{2}{2}} \textcolor{m a \ge n t a}{7} \text{ ; "color(blue)(6)" ; } \textcolor{b r o w n}{10} \textcolor{w h i t e}{\frac{2}{2}}\right\}$

$\left\{\textcolor{w h i t e}{\frac{2}{2}} \textcolor{m a \ge n t a}{7} \text{ ; "color(blue)(4.2->4 2/10)" ; } \textcolor{b r o w n}{4.9 \to 4 \frac{9}{10}} \textcolor{w h i t e}{\frac{2}{2}}\right\}$

#### Explanation:

Let the unknown sides of triangle B be b and c

The by ratio:

$\textcolor{b l u e}{\text{Condition 1}}$

$\frac{7}{36} = \frac{b}{42} = \frac{c}{60}$

$\implies$ The other two side lengths are:

$b = \frac{7 \times 42}{36} \approx 8.16 \overline{6}$ approximate value
$c = \frac{7 \times 60}{36} \approx 11.66 \overline{6}$ approximate value

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Condition 2}}$

$\frac{7}{42} = \frac{b}{36} = \frac{c}{60}$

$\implies$ The other two side lengths are:

$b = \frac{7 \times 36}{42} = 6$
$c = \frac{7 \times 60}{42} = 10$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Condition 3}}$

$\frac{7}{60} = \frac{b}{36} = \frac{c}{42}$

$b = \frac{7 \times 36}{60} = 4.2$
$c = \frac{7 \times 42}{60} = 4.9$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine exact values for condition 1}}$

$10 b \approx 81.66 \overline{6}$
$100 b \approx 816.66 \overline{6}$

$100 b - 10 b = 735$

$90 b = 735$

$b = \frac{735}{90} = 8 \frac{1}{6}$
,.......................................................

$\textcolor{w h i t e}{. .} c \approx \textcolor{w h i t e}{.} 11.66 \overline{6}$
$10 c \approx 116.66 \overline{6}$

$10 c - c = 105$

$9 c = 105$

$c = \frac{105}{9} = 11 \frac{2}{3}$