Similar Triangles and Polygons Practice Problems

To answer the question, we first need to understand what "similarity ratio" means.

In similar triangles, the ratio between the sides is constant.

In the statement, we do not have data on any of the sides.

However, a similarity ratio of 1 means that the sides are exactly the same size.

That is, the triangles are not only similar but also congruent.

In the drawing, you can clearly see that the triangles are of different sizes and, therefore, clearly the similarity ratio between them is not 1.

From the data in the drawing, it seems that angle M is equal to angle B

Also, angle A is an angle shared by both triangles ABC and AMN

That is, triangles ABC and AMN are similar respectively according to the angle-angle theorem.

According to the letters, the sides that are equal to each other are:

$\frac{AB}{AM}=\frac{BC}{MN}=\frac{AC}{AN}$

Now we can calculate the ratio between the sides of the given triangles:

$MN=3,BC=6$$\frac{6}{3}=2$

The square of the ratio between the sides is equal to the ratio between the areas of the parallelograms:

$3^2:4^2=9:16$

Let's look at the order of letters of the triangles that match each other and see the ratio of the sides.

We will write accordingly:

Triangle ABC is similar to triangle DFE

The order of similarity ratio will be:

$\frac{AB}{DF}=\frac{BC}{FE}=\frac{AC}{DE}$

Now let's insert the existing data we have in the diagram:

$\frac{8y}{9y}=\frac{7m}{FE}$

Let's reduce y and we get:

$\frac{8}{9}FE=7m$

$FE=\frac{9}{8}\times7m$

$FE=7\frac{7}{8}m$

To find the perimeter of the small triangle, we need to follow these steps:

• Step 1: Understand the ratio of areas and relate it to the ratio of corresponding side lengths.
• Step 2: Use the areas' ratio to find the side length ratio.
• Step 3: Calculate the perimeter of the small triangle using the side length ratio.

First, recall the relationship between the areas of similar triangles and their side lengths: if two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Thus, we have:

$\left(\frac{\text{Side of small triangle}}{\text{Side of large triangle}}\right)^2 = \frac{\text{Area of small triangle}}{\text{Area of large triangle}} = \frac{9}{100}$

Taking the square root of both sides gives us the ratio of the side lengths:

$\frac{\text{Side of small triangle}}{\text{Side of large triangle}} = \sqrt{\frac{9}{100}} = \frac{3}{10}$

This tells us that each side of the small triangle is $\frac{3}{10}$ of the corresponding side of the large triangle. Consequently, this ratio applies to the perimeters of the triangles too.

Given that the perimeter of the large triangle is 129 cm, the perimeter of the small triangle is:

$\text{Perimeter of small triangle} = \frac{3}{10} \times 129 = 38.7 \, \text{cm}$

Therefore, the solution to the problem is $\mathbf{38.7}$ cm.