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Many people enjoy collecting miniature versions of famous characters or historical structures. These tiny copies are scale models of the originals — in other words, the model and the original figure are similar. This lesson will cover the characteristics of similar figures and explain how to calculate their measurements based on a specific scale.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

## How Do the Dimensions of Similar Figures Change After a Scale?

Tadeo is interested in making miniature models of various objects. He wants to know more about how the dimensions of similar figures change depending on the scale. Consider the applet below, which displays two similar triangles.
Tadeo changes the length scale factor (lsf) and examines how the area of the blue triangle changes.
• What conclusions could Tadeo draw from the applet?
• How are the scale factor and the ratio between the areas of the triangles related?
Discussion

## Relations Between the Similar Polygons

Two polygons are said to be similar if their corresponding sides are proportional and their corresponding angles are congruent. Because of this, there is a relation between the perimeters of similar polygons.

Rule

## Perimeters of Similar Polygons Theorem

If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths.

Let and be the perimeters of and respectively. Let be the scale factor between corresponding side lengths. Then, based on the above diagram, the following relation holds true.

### Proof

Let and be two similar polygons with perimeters and respectively. By the definition of similar polygons, corresponding side lengths are proportional, with the scale factor as the common ratio.
If the equations are added together, the left-hand side of the resulting equation will give the perimeter of The right-hand side will give the scale factor times the perimeter of
Finally, the Perimeters of Similar Polygons Theorem is obtained by dividing both sides by .

Example

## Making a Miniature Model of a Basketball Court

Tadeo likes playing basketball. He decides to make a miniature model of a basketball court with a length of centimeters.

External credits: freepik

A standard basketball court has a length of meters and a width of meters.

a Calculate the scale factor in this model. Write the answer in the simplest fraction form.
b Calculate the perimeter of the miniature basketball court. Round the answer to the nearest integer.

### Hint

a The ratio of the length of the model basketball court to the actual length is equal to the length scale factor.
b If two polygons are similar, then the ratio of their perimeters is equal to their length scale factor.

### Solution

a The miniature model of the basketball court will be a scale model of a real court, so the model and its original can be thought of as similar polygons. Therefore, their side lengths will be proportional and equal to the scale factor.
Before the ratio can be calculated, both lengths need to have the same units of measure. The length of the real basketball court is meters and the length of the scale model is centimeters. Convert meters to centimeters to have same units of measure.
Next, calculate the ratio of the length of the model to the corresponding actual length in centimeters.
The scale factor of the model will be or
b To calculate the perimeter of the model, first find the perimeter of the real basketball court.
Recall that the perimeter of a rectangle is two times the sum of its length and width.
The perimeter of the real basketball court is meters. Since the miniature model of the court and the real court are similar rectangles, the ratio of their perimeters is equal to the ratio of their corresponding side lengths, or the scale factor.
The scale factor is Substitute the scale factor and the perimeter of the real basketball court into the equation and solve for
The perimeter of the model is about meters. Finally, convert it to centimeters.
Discussion

## Areas of Similar Figures

Like with perimeters, there is a relation between the areas of similar polygons.

If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Let and be similar figures, and and be their respective areas. The length scale factor between corresponding side lengths is Here, the following conditional statement holds true.

### Proof

The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.

The area of a rectangle is the product of its length and its width.

Area of Area of
By the definition of similar polygons, the corresponding side lengths are proportional and equal to the scale factor
The next step is to substitute the expressions for and into the formula for which represents the area of
Simplify right-hand side
Notice that the expression on the right-hand side is times the area of or This proof has shown that the ratio of the areas of the similar rectangles is equal to the square of the ratio of their corresponding side lengths. This ratio is also called the area scale factor.
Example

## Making a Miniature Model of a Spectator Stand

Now Tadeo wants to build a miniature spectator stand for his miniature basketball stadium. He plans to use the following stand as a model.
External credits: macrovector
When viewed from the side, the stand looks like a right triangle. Tadeo knows that its hypotenuse is meters long and its height is meters.
a Tadeo makes his scale model stand with a height of centimeters. Calculate the scale factor in this model. Write the answer in the simplest fraction form.
b Calculate the triangular side area of the miniature spectator stand.

### Hint

a The ratio of the hypotenuse of the model spectator stand to the actual hypotenuse is equal to the length scale factor.
b If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

### Solution

a The real spectator stand looks like a right triangle from a side perspective.
The scale model of the stand and the real stand have to be similar figures, so they have the same ratio between their corresponding side lengths. This ratio is called the scale factor.
Tadeo wants to create the model with a centimeter height. He knows that the height of the real stand is meters. Before finding the scale factor, both lengths must be in the same units. To achieve this, convert the height of the real stands from meters to centimeters.
Now the ratio of the height of the model to the corresponding actual height can be calculated.
The scale factor of the model will be or This means that the dimensions of the model stand will be times smaller than the dimensions of the real stand.
b Before calculating the side area of the model spectator stand, calculate the side area of the actual stand. This requires finding the length of the other leg.
Since it is a right triangle, the Pythagorean theorem can be used to find
Solve for
The base length of the triangular figure is meters. Now, calculate the area of the triangle as half of the product of the base and height.
The area of the triangular side of the real stand is square meters. Recall that if two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
The ratio of the corresponding side lengths is equal to the length scale factor, so in this case, The square of this scale factor can be used to find the area of the model's triangular side.
Solve for
The area of the model's triangular side is square meters. Finally, convert it to square centimeters.
The area of the triangular side of Tadeo's model is square centimeters.
Pop Quiz

## Finding the Length Scale, Perimeter, or Area of Similar Polygons

Consider two similar polygons. Use the given information to find the scale factor, perimeter, or area of either of the polygons. Keep in mind that the given length scale factor corresponds to the ratio of Polygon to Polygon Round the answer to two decimal places if necessary.