Given two similar triangles with a scale factor of a : b, show the ratio of their perimeters is also a : b?

1 Answer
sente
Sep 17, 2016

Suppose two triangles #triangleA# and #triangleB# are similar with a scale factor of #S = a/b#. Then, for each side #bar(A_i), i = 1, 2, 3# of triangle #A#, triangle #B# has a corresponding side #bar(B_i)# such that #A_i = SB_i#.

As the perimeter of a triangle is equal to the sum of the lengths of its sides, we have the perimeter #P_A# of #triangleA# as

#P_A = A_1+A_2 + A_3#

#=SB_1+SB_2+SB_3#

#=S(B_1+B_2+B_3)#

#=SP_B#

Thus the ratio of the perimeters is the same as the ratio of the sides, that is, of the scale factor between the triangles.

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