Question

The widths of two similar triangles are 12 m and 8 m. What is the ratio of the perimeters? Of the areas?

Answer 1

Answer:

Ratio of Perimeters = 3 : 2

Ratio of Areas = 9 : 4

Step-by-step explanation:

Ratio of Perimeters Between Similar Triangles

The ratio of similarity between two similar triangles (the ratio between every two similar sides between similar triangles) is equal to the ratio between the triangles' parameters.

To prove this, let there be triangle A with sides x, y and z. We'll construct another triangle, triangle B, that is similar to triangle A and whose sides are k times larger than triangle A's sides. Thus, triangle B's sides are kx, ky and kz.

The ratio of similarity can be found by dividing any side of triangle A by its similar side in triangle B.

$\text{Ratio of similarity} = \frac{x}{kx} = \frac{y}{ky} = \frac{z}{kz} = \frac1k$

The ratio of perimeters can be found by finding each of the triangles' perimeters and dividing the perimeter of triangle A by that of triangle B.

$P_A = x + y + z\\P_B = kx + ky + kz \text{ // Take out common factor }k\\\to P_B = k(x + y + z)\\\\\text{Ratio of Perimeters} = \frac{P_A}{P_B} = \frac{x + y + z}{k(x + y + z)} = \frac1k$

Thus, the ratio of similarity is equal to the ratio of parameters.

If the two widths of the similar triangles are 12 and 8, then the ratio of similarity is 12/8 which is 3/2. Thus, the ratio of the perimeters is also 3/2.

Ratio of Areas Between Similar Triangles

The ratio of areas between two similar triangles is equal to the ratio of similarity squared.

We can show this statement is true by finding the ratio of similarity between two similar triangles and the ratio between their areas, and showing that the latter is equal to the first squared.

To keep it short, the triangles I will use for this proof are the same as the triangles in the previous proof with the same ratio of similarity, 1/k.

For this proof, I'll also define another variable, h, being triangle A's height to base x. Thus, triangle B's height to base kx is kh.

$S_A = \frac{x \times h}2 = \frac{xh}2\\S_B = \frac{kx \times kh}2 \text{ // Simplify}\\\to S_B = \frac{k^2xh}2\\\\\text{Ratio of Areas} = \frac{S_A}{S_B} = \frac{\frac{xh}2}{\frac{k^2xh}2} = \frac1{k^2} \times \frac{\frac{xh}2}{\frac{xh}2} = \frac1{k^2} = \left(\frac1k\right)^2 = (\text{Ratio of Similarity})^2$

Thus, the ratio of areas is equal to the ratio of similarity squared.

We have shown previously that the ratio of similarity is 3/2, thus the ratio of areas is (3/2)^2 = 9/4.