Triangle A has sides of lengths #36 #, #44 #, and #32 #. Triangle B is similar to triangle A and has a side of length #4 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #36    #, #44    #, and #32     #. Triangle B is similar to triangle A and has a side of length #4     #. What are the possible lengths of the other two sides of triangle B?

Autumn Armstrong · Accepted Answer

Possible lengths of other two sides of triangle B are

Case 1 : #4.8889, 3.5556#

Case 2 : #3.2727, 2.9091#

Case 3 : #4.5, 5.5# Let the sides be a1, a2, a3 of #Delta#A and b1, b2, b3 of #Delta# B. We know, #(a1) / (b1) = (a2) / (b2) = (a3) / (b3)# Given #a1 = 36, a2 = 44, a3 = 32# Case 1 : #b1 = 4# Then #b2 = ((a2) * (b1)) / (a1) = (44*4)/36 = 4.8889# #b3 = ((a3) * (b1)) / (a1) = (32 * 4) / 36 = 3.5556# Case 2 : #b2 = 4# #b1 = ((a1)*(b2)) / (a2) = (36*4)/44 = 3.2727# #b3 = (32 * 4) / 44 = 2.9091# Case 3 : #b3 = 4# #b1 = (36 * 4) / 32 = 4.5# #b2 = (44 * 4) / 32 = 5.5#

Wyatt Anderson · Answer

undefined The possible lengths of the other two sides of triangle B, given that it is similar to triangle A and has a side of length 4, can be found by setting up proportions based on the similarity of the triangles. Since the triangles are similar, corresponding sides are in proportion.

Let $ x $ represent one of the unknown sides of triangle B. Using the side lengths of the triangles, we can set up the following proportion:

$$
\frac{36}{x} = \frac{44}{4} = \frac{32}{y}
$$

Solving for $ y $ gives:

$$
y = \frac{32 \times 4}{44} = \frac{128}{44}
$$

So, the possible lengths of the other two sides of triangle B are $ x $ and $ \frac{128}{44} $.