The sides triangle are 8, 12, and 15. The longest side of a similar triangle is 18. What is the ratio of perimeter of the smaller triangle to the perimeter of the larger triangle?

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Answer 1 · 2018-05-30T16:20:20

#color(blue)(5:6)#

For similar triangles:

#DeltaABC~DeltaDEF# we have:

#(AB)/(DE)=(BC)/(EF)=(AC)/(DF)#

The perimeter of a triangle is a linear measurement, therefore the perimeters will be in the same ratio as the sides:

Ratio of the longest side of smaller triangle to the longest side of larger triangle:

#15:18=5:6#

Ratio of perimeters is the same:

#5:6#

Any two corresponding sides could have been used, but we were given the two longest, so we used them.

We can show this is correct by finding the similar triangle:

#Delta ABC# has sides:

#AB=15# , #BC=12#, #AC=8#

#Delta DEF# has sides:

#DE=18# , given.

#EF# , #DF#

So by the property of similar triangles:

#(AB)/(DE)=(BC)/(EF)=(AC)/(DF)#

#15/18=12/(EF)=>EF=(12*18)/15=72/5#

#15/18=8/(DF)=>DF=(8*18)/15=48/5#

Similar triangle has sides:

#DE=18# , #EF=72/5# , #DF=48/5#

Perimeter of ABC:

#15+12+8=35#

Perimeter of DEF:

#18+72/5+48/5=42#

Ratio of smaller to larger:

#35:42=5:6#

This is what we expected.