# Lesson 1

Scale Drawings

• Let’s make a scale drawing.

### 1.1: Is That the Same Hippo?

Diego took a picture of a hippo and then edited it. Which is the distorted image? How can you tell?

Is there anything about the pictures you could measure to test whether there’s been a distortion?

### 1.2: Sketching Stretching

A dilation with center $$O$$ and positive scale factor $$r$$ takes a point $$P$$ along the ray $$OP$$ to another point whose distance is $$r$$ times farther away from $$O$$ than $$P$$ is. If $$r$$ is less than 1 then the new point is really closer to $$O$$, not farther away.

1. Dilate $$H$$ using $$C$$ as the center and a scale factor of 3. $$H$$ is 40 mm from $$C$$.
2. Dilate $$K$$ using $$O$$ as the center and a scale factor of $$\frac{3}{4}$$. $$K$$ is 40 mm from $$O$$.

### 1.3: Mini Me

1. Dilate the figure using center $$P$$ and scale factor $$\frac12$$.

2. What do you notice? What do you wonder?

1. Dilate segment $$AB$$ using center $$P$$ by scale factor $$\frac12$$. Label the result $$A'B'$$.
2. Dilate the segment $$AB$$ using center $$Q$$ by scale factor $$\frac12$$.
3. How does the length of $$A''B''$$ compare to $$A'B$$? How would the length of $$A''B''$$ change if $$Q$$ was infinitely far away? Explain or show your answer.

### Summary

A scale drawing of an object is a drawing in which all lengths in the drawing correspond to lengths in the object by the same scale. When we scale a figure we need to be sure to scale all of the parts equally or else the image will become distorted.

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor. The scale factor is the factor by which every length in a original figure is multiplied when you make a scaled copy. A scale factor greater than 1 enlarges an object while a scale factor less than 1 shrinks an object. What would a scale factor equal to 1 do?

For example, segment $$BC$$ is a scaled copy of segment $$DE$$ with a scale factor of $$\frac14$$. So $$BC=\frac14DE$$. If $$DE=6$$, then $$BC=\frac64$$ or 1.5.

To perform a dilation, we need a center of dilation, a scale factor, and something to dilate. A dilation with center $$A$$ and positive scale factor $$k$$ takes a point $$D$$ along the ray $$AD$$ to another point whose distance is $$k$$ times farther away from $$A$$ than $$D$$ is.

Segment $$FG$$ is a dilation of segment $$DE$$ using center $$A$$ and a scale factor of 3. So $$FA=3 \boldcdot DA$$. If $$DA=15$$, then $$FA=45$$.

### Glossary Entries

• dilation

A dilation with center $$P$$ and positive scale factor $$k$$ takes a point $$A$$ along the ray $$PA$$ to another point whose distance is $$k$$ times farther away from $$P$$ than $$A$$ is.

Triangle $$A'B'C'$$ is the result of applying a dilation with center $$P$$ and scale factor 3 to triangle $$ABC$$.

• scale factor

The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.