Triangle A has sides of lengths #42 ,36 #, and #21 #. Triangle B is similar to triangle A and has a side of length #14 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #42  ,36  #, and #21 #. Triangle B is similar to triangle A and has a side of length #14   #. What are the possible lengths of the other two sides of triangle B?

Aurora Ayala · Accepted Answer

The possible length of sides for triangle B are #{14,12,7}#, #{14,49/3,49/6}#,#{14,28,24}#

Let say 14 is a length of triangle B reflect to the length of 42 for triangle A and X,Y are the length for other two sides of triangle B.

#X/36 = 14/42# #X=14/42*36# #X=12#

#Y/21 = 14/42# #Y=14/42*21# #Y=7# The length of sides for triangle B are #{14,12,7}#

Let say 14 is a length of triangle B reflect to the length of 36 for triangle A and X,Y are the length for other two sides of triangle B. #X/42 = 14/36# #X=14/36*42# #X=49/3#

#Y/21 = 14/36# #Y=14/36*21# #Y=49/6# The length of sides for triangle B are #{14,49/3,49/6}#

Let say 14 is a length of triangle B reflect to the length of 21 for triangle A and X,Y are the length for other two sides of triangle B. #X/42 = 14/21# #X=14/21*42# #X=28#

#Y/36 = 14/21# #Y=14/21*36# #Y=24# The length of sides for triangle B are #{14,28,24}#

Savannah Aguilar · Answer

undefined Since triangles A and B are similar, their corresponding sides are proportional. Let's denote the lengths of the sides of triangle B as $ x $ and $ y $, where $ x $ is the side corresponding to 36 in triangle A, and $ y $ is the side corresponding to 21 in triangle A.

Using the property of similar triangles, we can set up the following proportion:

$$
\frac{x}{42} = \frac{14}{36}
$$

$$
\frac{y}{42} = \frac{14}{21}
$$

Solving these proportions for $ x $ and $ y $:

$$
x = \frac{14 \times 42}{36}
$$

$$
y = \frac{14 \times 42}{21}
$$

Calculating these values:

$$
x = \frac{14 \times 42}{36} = \frac{14 \times 7}{6} = \frac{98}{6} = \frac{49}{3}
$$

$$
y = \frac{14 \times 42}{21} = 14 \times 2 = 28
$$

So, the possible lengths of the other two sides of triangle B are $ \frac{49}{3} $ and 28.