# Triangle A has sides of lengths 60 , 42 , and 54 . 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?

Jan 28, 2018

Possible lengths of two triangles are

Case 1 : color(green)(A (42, 54, 60) & B (7. 8.2727, 10))

Case 2 : color(brown)(A (42, 54, 60) & B (5.4444, 7, 7.7778))

Case 3 : color(blue)( A (42, 54, 60) & B (4.9, 6.3, 7))

#### Explanation:

Let the two triangles A & B have sides PQR & XYZ respectively.

$\frac{P Q}{X Y} = \frac{Q R}{Y Z} = \frac{R P}{Z X}$

Case 1 : Let XY = $\textcolor{g r e e n}{7}$

$\frac{42}{7} = \frac{54}{Y Z} = \frac{60}{Z X}$

$Y Z = \frac{54 \cdot 7}{42} = \textcolor{g r e e n}{8.2727}$

$Z X = \frac{60 \cdot 7}{42} = \textcolor{g r e e n}{10}$

Case 2 : Let YZ = $\textcolor{b r o w n}{7}$

$\frac{42}{X Y} = \frac{54}{7} = \frac{60}{Z X}$

$X Y = \frac{42 \cdot 7}{54} = \textcolor{b r o w n}{5.4444}$

$Z X = \frac{60 \cdot 7}{54} = \textcolor{b r o w n}{7.7778}$

Case 3 : Let ZX = $\textcolor{b l u e}{7}$

$\frac{42}{X Y} = \frac{54}{Y} Z = \frac{60}{7}$

$X Y = \frac{42 \cdot 7}{60} = \textcolor{b l u e}{4.9}$

$Y Z = \frac{54 \cdot 7}{60} = \textcolor{b l u e}{6.3}$