The ratio of the perimeters of two similar triangles is 1:3. The area of A larger triangle is 27ft square. What is the area of the smallest triangle?

Question

The scale factor of triangle ABC to triangle DEF is 3/5

Answer 1

When two triangles are similar the ratio of their areas is the square of the ratio of their perimeters, so if the ratio of their perimeters is 1:3, the ratio of their areas is 1:9.

We are given the area of the larger triangle is 27 ft and can now use ratio of their areas to find the area of the smaller triangle.

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Answer 2

Here are two ways of solving the problem.

In similar triangles, the ratio of their areas equals the square of the ratio of their sides.

The ratio of their perimeters is 1:3 so the ratio of their sides and their altitudes will also be in the ratio of 1:3.

That given, the ratios of the areas is 1^2/3^2=1/9. (ratio of their sides: a/b, ratio of their areas: a^2/b^2)

Therefore we have 1/9=A/27 where A=area of the smaller triangle

cross multiply

1*27=9*A

27=9A

A=27/9

A=3 square feet is the area of the smaller triangle

Suppose that you did not know this.

Let one triangle have sides a, b and c.

Let the other triangle be 3a, 3b and 3c.

Let h=altitude of the smaller triangle.

3h=altitude of the larger triangle

let b and 3b be the bases of the triangles

A=(1/2)(b)(h) is the area of the smaller triangle.

A=(1/2)(3b)(3h) is the area of the larger triangle.

27=(1/2)(9bh)

27=4.5bh

bh=27/4.5

bh=6

substitute this into the first area formula for the smaller triangle

A=(1/2)(6)

A=3 square feet is the area of the smaller triangle again