Triangle A has sides of lengths #36 #, #42 #, and #60 #. 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?

Question

Triangle A has sides of lengths #36    #, #42    #, and #60     #. 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?

Savannah Aguilar · Accepted Answer

#{color(white)(2/2)color(magenta)(7)" ; "color(blue)(8.16bar6-> 8 1/6)" ; "color(brown)(11.6bar6->11 2/3)color(white)(2/2)}#

#{color(white)(2/2)color(magenta)(7)" ; "color(blue)(6)" ; "color(brown)(10)color(white)(2/2)}#

#{color(white)(2/2)color(magenta)(7)" ; "color(blue)(8.16bar6-> 8 1/6)" ; "color(brown)(11.6bar6->11 2/3)color(white)(2/2)}#

#{color(white)(2/2)color(magenta)(7)" ; "color(blue)(6)" ; "color(brown)(10)color(white)(2/2)}#

#{color(white)(2/2)color(magenta)(7)" ; "color(blue)(4.2->4 2/10)" ; "color(brown)(4.9->4 9/10)color(white)(2/2)}#

Let the unknown sides of triangle B be b and c

The by ratio:

#color(blue)("Condition 1")#

#7/36=b/42=c/60#

#=># The other two side lengths are:

#b=(7xx42)/36 ~~ 8.16bar6# approximate value #c=(7xx60)/36~~11.66bar6# approximate value

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Condition 2")#

#7/42=b/36=c/60#

#=># The other two side lengths are:

#b=(7xx36)/42=6# #c=(7xx60)/42=10# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Condition 3")#

#7/60=b/36=c/42#

#b=(7xx36)/60=4.2# #c=(7xx42)/60=4.9# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine exact values for condition 1")#

#10b~~81.66bar6# #100b~~816.66bar6#

#100b-10b=735#

#90b=735#

#b=735/90=8 1/6# ,.......................................................

#color(white)(..)c~~color(white)(.)11.66bar6# #10c~~116.66bar6#

#10c-c=105#

#9c=105#

#c=105/9=11 2/3#

Sophia Alfred · Answer

undefined Using the property of similar triangles, the ratios of corresponding sides of similar triangles are equal.

Let the sides of triangle B be $ x $ and $ y $, where $ x $ corresponds to the side of length 36 in triangle A and $ y $ corresponds to the side of length 42 in triangle A.

Thus, we have the following ratios:

$$ \frac{x}{36} = \frac{7}{42} $$
$$ \frac{y}{42} = \frac{7}{42} $$

Solving for $ x $ and $ y $, we find:

$$ x = \frac{7}{6} \times 36 = 42 $$
$$ y = \frac{7}{6} \times 42 = 49 $$

Therefore, the possible lengths of the other two sides of triangle B are 42 and 49.