# Lesson 5

Scaling and Unscaling

• Let’s examine the relationships between areas of dilated figures and scale factors.

### 5.1: Transamerica Building

The image shows the Transamerica Building in San Francisco. It’s shaped like a pyramid.

The bottom floor of the building is a rectangle measuring approximately 53 meters by 44 meters. The top floor of the building is a dilation of the base by scale factor $$k=0.32$$.

Ignoring the triangular “wings” on the sides, what is the area of the top floor? Explain or show your reasoning.

### 5.2: Two Viewpoints

A triangle has area 100 square inches. It’s dilated by a factor of $$k=0.25$$.

Mai says, “The dilated triangle’s area is 25 square inches.”

Lin says, “The dilated triangle’s area is 6.25 square inches.”

1. For each student, decide whether you agree with their statement. If you agree, explain why. If you disagree, explain what the student may have done to arrive at their answer.
2. Calculate the area of the image if the original triangle is dilated by each of these scale factors:
1. $$k=9$$
2. $$k=\frac34$$

### 5.3: Graphing Areas and Scale Factors

An artist painted a 1 foot square painting. Now she wants to create more paintings of different sizes that are all scaled copies of her original painting. The paint she uses is expensive, so she wants to know the sizes she can create using different amounts of paint.

1. Suppose the artist has enough paint to cover 9 square feet. If she uses all her paint, by what scale factor can she dilate her original painting?
2. Complete the table that shows the relationship between the dilated area ($$x$$) and the scale factor ($$y$$). Round values to the nearest tenth if needed. Use the applet at the end of the lesson to help, if you choose.
dilated area in square feet scale factor
0
1
4
9
16
3. On graph paper, plot the points from the table and connect them with a smooth curve.
4. Use your graph to estimate the scale factor the artist could use if she had enough paint to cover 12 square feet.
5. Suppose the painter has enough paint to cover 1 square foot, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?
6. Suppose the painter has enough paint to cover 14 square feet, and she buys enough paint to cover an additional 2 square feet. How does this change the scale factor she can use?

Use the applet to help, if you choose.

The image shows triangle $$ABC$$.

1. Sketch the result of dilating triangle $$ABC$$ using a scale factor of 2 and a center of $$A$$. Label it $$AB'C'$$.
2. Sketch the result of dilating triangle $$ABC$$ using a scale factor of -2 and a center of $$A$$. Label it $$AB''C''$$.
3. Find a transformation that would take triangle $$AB'C'$$ to $$AB''C''$$.

### Summary

If we know the area of an original figure and its dilation, we can work backwards to find the scale factor. For example, suppose we have a circle with area 1 square unit, and a dilation of the circle with area 64 square units. We know the circle must have been dilated by a factor of 8, because 82 = 64. Another way to say this is $$\sqrt{64}=8$$.

A graph can help us understand the relationship between dilated areas and scale factors. We can make a table of values for the dilated circle, plot the points on a graph, and connect them with a smooth curve. In this table, the dilated area is the input or $$x$$ value, and the scale factor is the output or $$y$$ value. Remember that the area of the original circle is 1 square unit, so the square root of the dilated area is the same as the scale factor.

dilated area in square units scale factor
0 0
1 1
4 2
9 3
16 4

This graph represents the equation that describes the relationship between area and scale factor: $$y=\sqrt{x}$$. Note that the rate of change isn’t constant. On the left side, the graph is fairly steep. As the area increases, the scale factor increases quickly. But on the right side, the graph flattens out. As the area continues to increase, the scale factor still increases, but not as quickly.

### Glossary Entries

• axis of rotation

A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.

• cone

A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.

• cross section

The figure formed by intersecting a solid with a plane.

• cylinder

A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.

• face

Any flat surface on a three-dimensional figure is a face.

A cube has 6 faces.

• prism

A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”

• pyramid

A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”

• solid of rotation

A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.

The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.

• sphere

A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.