Triangle A has sides of lengths #54 #, #44 #, and #64 #. 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 #54    #, #44    #, and #64     #. 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?

Evelyn Arboleda · Accepted Answer

#(8,176/27,256/27) , (108/11,8,128/11) , (27/4,11/2,8)#

Since the triangles are similar then the ratios of corresponding sides are equal.

Name the 3 sides of triangle B , a , b and c , corresponding to the sides 54 , 44 and 64 in triangle A. #"------------------------------------------------------------------------"#

If side a = 8 then ratio of corresponding sides = #8/54 = 4/27 #

Hence b = # 44xx4/27 = 176/27" and " c = 64xx4/27 = 256/27 #

The 3 sides in B # = (8,176/27,256/27 ) # #"------------------------------------------------------------------------"#

If side b = 8 then ratio of corresponding sides# = 8/44 = 2/11 #

hence a = # 54xx2/11 = 108/11" and " c = 64xx2/11 = 128/11 #

The 3 sides in B = #(108/11,8,128/11 )# #"------------------------------------------------------------------------"#

If side c = 8 then ratio of corresponding sides #= 8/64 = 1/8 #

hence a #=54xx1/8 = 27/4" and " b = 44xx1/8 = 11/2 #

The 3 sides in B =# (27/4,11/2,8 )# #"-----------------------------------------------------------------------------"#

Maverick Smith · Answer

undefined Since triangle B is similar to triangle A, the ratio of corresponding sides of triangles A and B must be the same. Let x be the length of one of the other two sides of triangle B. Using the ratio of corresponding sides:

$$ \frac{x}{54} = \frac{8}{44} $$

Solving for x gives:

$$ x = \frac{8 \times 54}{44} = \frac{432}{44} = 9 $$

Therefore, one of the other two sides of triangle B is 9 units long.

Using the same ratio, we can find the length of the third side:

$$ \frac{x}{64} = \frac{8}{44} $$

Solving for x gives:

$$ x = \frac{8 \times 64}{44} = \frac{512}{44} = 11.64 $$

Therefore, the possible lengths of the other two sides of triangle B are approximately 9 units and 11.64 units.