Unit 1 Scale Drawings — Unit Plan

What is a scaled copy of a figure? Let’s look at some examples.

The second and third drawings are both scaled copies of the original Y.

However, here, the second and third drawings are not scaled copies of the original W.

3 drawings of the letter W. The second drawing is spread out making it wider and shorter than the first drawing. The third drawing is squished in, making it narrower, but the same height as the the original drawing.

The second drawing is spread out (wider and shorter). The third drawing is squished in (narrower, but the same height).

We will learn more about what it means for one figure to be a scaled copy of another in upcoming lessons.

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.

For example, to make a scaled copy of triangle $ABC$ where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle $DEF$, each side is 4 times as long as the corresponding side in triangle $ABC$.

The size of the scale factor affects the size of the copy. When a figure is scaled by a scale factor greater than 1, the copy is larger than the original. When the scale factor is less than 1, the copy is smaller. When the scale factor is exactly 1, the copy is the same size as the original.

Triangle $DEF$ is a larger scaled copy of triangle $ABC$, because the scale factor from $ABC$ to $DEF$ is $\frac32$. Triangle $ABC$ is a smaller scaled copy of triangle $DEF$, because the scale factor from $DEF$ to $ABC$ is $\frac23$.

Two triangles; one labeled A B C with horizontal A B and the other D E F with horizontal D E. The length of A B is labeled 4. The length of B C is labeled 3. The length of C A is labeled 5. The length of D E is labeled 6. The length of E F is labeled 4.5. The length of F D is labeled 7.5. An arrow from triangle A B C pointing to triangle D E F is labeled, times 3 halves. An arrow from triangle D E F pointing to triangle A B C is labeled times 2 thirds.

This means that triangles $ABC$ and $DEF$ are scaled copies of each other. It also shows that scaling can be reversed using reciprocal scale factors, such as $\frac23$ and $\frac32$.

In other words, if we scale Figure A using a scale factor of 4 to create Figure B, we can scale Figure B using the reciprocal scale factor, $\frac14$, to create Figure A.

Scaling affects lengths and areas differently. When we make a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3, the side lengths of the copy will be 6 units and 12 units, because $2\boldcdot 3 = 6$ and $4\boldcdot 3 = 12$.

The area of the copy, however, changes by a factor of (scale factor)². If each side length of the copy is 3 times longer than the original side length, then the area of the copy will be 9 times the area of the original, because $3\boldcdot 3$, or $3^2$, equals 9.

Two rectangles. The first rectangle has the vertical side labeled 2 and the horizontal side labeled 4. The second rectangle has the vertical side labeled 6 and the horizontal side labeled 12. Two horizontal dashed lines and 2 vertical dashed lines are drawn in the second rectangle dividing it into 9 identical smaller rectangles.

In this example, the area of the original rectangle is 8 units² and the area of the scaled copy is 72 units², because $9\boldcdot 8 = 72$. We can see that the large rectangle is covered by 9 copies of the small rectangle, without gaps or overlaps. We can also verify this by multiplying the side lengths of the large rectangle: $6\boldcdot 12=72$.

Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two‑dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length $l$ and width $w$. If we scale the rectangle by a scale factor of $s$, we get a rectangle with length $s\boldcdot l$ and width $s\boldcdot w$. The area of the scaled rectangle is $A = (s\boldcdot l) \boldcdot (s\boldcdot w)$, so $A= (s^2) \boldcdot (l \boldcdot w)$. The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.

Scale drawings are two-dimensional representations of actual objects or places. Floor plans and maps are some examples of scale drawings. On a scale drawing:

• Every part corresponds to something in the actual object.
• Lengths on the drawing are enlarged or reduced by the same scale factor.
• A scale tells us how actual measurements are represented on the drawing. For example, if a map has a scale of “1 inch to 5 miles” then a $\frac12$-inch line segment on that map would represent an actual distance of 2.5 miles.

Sometimes the scale is shown as a segment on the drawing itself. For example, here is a scale drawing of a stop sign with a line segment that represents 25 cm of actual length.

The width of the octagon in the drawing is about three times the length of this segment, so the actual width of the sign is about $3 \boldcdot 25$, or 75 cm.

Because a scale drawing is two-dimensional, some aspects of the three-dimensional object are not represented. For example, this scale drawing does not show the thickness of the stop sign.

A scale drawing may not show every detail of the actual object; however, the features that are shown correspond to the actual object and follow the specified scale.

Maps with scales are useful for making calculations involving speed, time, and distance. Here is a map of part of Alabama.

A map of a part of Alabama. Three points indicate three different cities on the map. The bottom point is labeled Montgomery, the middle point is labeled Centreville, and the top point is labeled Birmingham. A scale on the map shows 1 inch equals 20 miles. The distance between Birmingham and Montgomery is approximately 4 point 5 inches on the map.

Suppose it takes a car 1 hour and 30 minutes to travel at constant speed from Birmingham to Montgomery. How fast is the car traveling?

To make an estimate, we need to know about how far it is from Birmingham to Montgomery. The scale of the map represents 20 mi, so we can estimate that the distance between these cities is about 90 mi.

Table with arrows. Time, hours. Data, from top to bottom, 1.5, 3, 1. Distance, miles. Data from top to bottom, 90, 180, 60. On the left and right of the table, two downward arrows labeled times 2 and times one third. 

Since 90 miles in 1.5 hours is the same speed as 180 mi in 3 hours, the car is traveling about 60 mi per hour.

Suppose a car is traveling at a constant speed of 60 miles per hour from Montgomery to Centreville. How long will it take the car to make the trip? Using the scale, we can estimate that it is about 70 mi. Since 60 miles per hour is the same as 1 mile per minute, it will take the car about 70 minutes (or 1 hour and 10 minutes) to make this trip.

If we want to create a scale drawing of a room's floor plan that has the scale “1 inch to 4 feet,” we can divide the actual lengths in the room (in feet) by 4 to find the corresponding lengths (in inches) for our drawing.

Suppose the longest wall is 23 feet long. We should draw a line 5.75 inches long to represent this wall, because $23 \div 4 = 5.75$.

Any of these scales can be used to find actual lengths and scaled lengths (lengths on a drawing). For instance, we can tell that, at this scale, an 8-foot long wall should be 2 inches long on the drawing because $\frac14 \boldcdot 8 = 2$.

The size of a scale drawing is influenced by the choice of scale. For example, here is another scale drawing of the same room using the scale 1 inch to 8 feet.

Notice that this drawing is smaller than the previous one. Since one inch on this drawing represents twice as much actual distance, each side length needs to be only half as long as it was in the first scale drawing.