Triangle A has sides of lengths #32 #, #40 #, and #16 #. Triangle B is similar to triangle A and has a side of length #8 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #32    #, #40    #, and #16     #. Triangle B is similar to triangle A and has a side of length #8     #. What are the possible lengths of the other two sides of triangle B?

Maverick Smith · Accepted Answer

Three possible lengths of other two sides are #(16,20) or(4,10) or (3.2,6.4)# unit each

The ratio of sides of triangle A is # 16/8:32/8:40/8 or 2:4:5# The ratio of sides of similar triangle B must have the same ratio. If #8# be the lowest side then other two sides are # 4*8/2=16 and 5*8/2=20# unit If #8# be the middle one then other two sides are # 2*8/4=4 and 5*8/4=10# unit if #8# be the biggest side then other two sides are # 2*8/5=3.2 and 4*8/5=6.4#unit Hence three possible lengths of other two sides are #(16,20) or (4,10),or (3.2,6.4)# unit each [Ans]

Sadie Adams · Answer

undefined Since triangle B is similar to triangle A, the ratios of corresponding sides in the two triangles are equal. Let's denote the lengths of the corresponding sides of triangles A and B as follows:

$ \frac{{\text{Side of Triangle B}}}{{\text{Side of Triangle A}}} = \frac{{8}}{{32}} = \frac{{\text{length of corresponding side in B}}}{{\text{length of corresponding side in A}}} $

$ \frac{{\text{Side of Triangle B}}}{{\text{Side of Triangle A}}} = \frac{{8}}{{40}} = \frac{{\text{length of corresponding side in B}}}{{\text{length of corresponding side in A}}} $

$ \frac{{\text{Side of Triangle B}}}{{\text{Side of Triangle A}}} = \frac{{8}}{{16}} = \frac{{\text{length of corresponding side in B}}}{{\text{length of corresponding side in A}}} $

Now, we can find the lengths of the other two sides of triangle B by multiplying the corresponding lengths of triangle A by the ratios calculated above.

For the first ratio $ \frac{{8}}{{32}} $, we find:

$ \frac{{8}}{{32}} \times 40 = 10 $

For the second ratio $ \frac{{8}}{{40}} $, we find:

$ \frac{{8}}{{40}} \times 32 = 6.4 $

For the third ratio $ \frac{{8}}{{16}} $, we find:

$ \frac{{8}}{{16}} \times 40 = 20 $

So, the possible lengths of the other two sides of triangle B are 10, 6.4, and 20.