The ratio of the areas of two
similar triangles is nine 100th. If the perimeter of the larger
triangle is 129, what is the perimeter of the smaller one?
Notice that we’re given the ratio
of the areas. But we’re asked to solve for a
perimeter. That means we do not want to use
the ratio of an area. We want to use the ratio of a
perimeter. The relationship between an area
ratio and a perimeter ratio is that an area ratio is equal to a perimeter ratio
Essentially, the ratio of the areas
of the surfaces of two similar triangles equals the square of the ratio of the
length of any two corresponding sides of the triangle. And since the ratio of the area of
two similar triangles we know to be equal to nine 100th, let’s go ahead and plug
that in for the area ratio. So here, we’ve done that.
And as we said before, the ratio of
the areas of the surfaces of two similar triangles equals the square of the ratios
of the lengths. Well, a perimeter is a length. So what we have is perfect. So in order to solve for a
perimeter ratio, we need to square root both sides of the equation. So we need to take the square root
of nine and the square root of 100. The square root of nine is three
and the square root of 100 is 10.
So now, let’s use this perimeter
ratio to solve for the perimeter of the smaller triangle. So out of three 10ths, three is the
smaller number. So it needs to be across from the
smaller perimeter. And since 10 is the larger number,
it should be across from the larger perimeter. And we actually know the perimeter
of the larger triangle. So we will plug in 129 for the
larger triangle perimeter.
And so to solve for the smaller
perimeter, we need to cross multiply and solve. So we take three times 129 and set
it equal to the smaller perimeter times 10. So three times 129 is equal to
387. Now, we need to divide both sides
by 10, resulting in 38.7 will be the perimeter of the smaller triangle.