Chapter Notes: Scale Copies
Imagine you have a photograph of your family and you want to make it bigger to frame on the wall, or smaller to fit in your wallet. When you resize the photo, you want it to look exactly the same-just larger or smaller. The people in the photo shouldn't look stretched or squished. In mathematics, when we create a larger or smaller version of a figure that looks exactly the same, we call it a scale copy. Scale copies are all around us: maps, blueprints, models of buildings, and even when you zoom in or out on your phone screen. Understanding scale copies helps us work with sizes and measurements in the real world.
What Is a Scale Copy?
A scale copy is a new figure that has the same shape as the original figure but may be a different size. The key idea is that every length in the scale copy is multiplied by the same number, called the scale factor.
When we make a scale copy, two important things must be true:
- All corresponding angles stay the same size
- All corresponding lengths are multiplied by the same scale factor
If a figure is a scale copy of another, we say the two figures are similar. Similar figures have the same shape but not necessarily the same size.
The Scale Factor
The scale factor is the number we multiply each length by to create the scale copy. If the scale factor is greater than 1, the scale copy is larger than the original. If the scale factor is less than 1 (but greater than 0), the scale copy is smaller than the original. If the scale factor equals 1, the scale copy is exactly the same size as the original.
We can write the relationship like this:
\[ \text{Length in scale copy} = \text{Scale factor} \times \text{Length in original} \]Where the scale factor is a positive number.
Example: A rectangle has a length of 6 cm and a width of 4 cm.
You create a scale copy using a scale factor of 2.What are the dimensions of the scale copy?
Solution:
Multiply the original length by the scale factor:
New length = 2 × 6 = 12 cmMultiply the original width by the scale factor:
New width = 2 × 4 = 8 cmThe scale copy is a rectangle with length 12 cm and width 8 cm.
Example: A triangle has sides measuring 9 inches, 12 inches, and 15 inches.
You create a scale copy using a scale factor of 1/3.What are the side lengths of the scale copy?
Solution:
Multiply each side length by the scale factor 1/3:
First side: \( \frac{1}{3} \times 9 = 3 \) inches
Second side: \( \frac{1}{3} \times 12 = 4 \) inches
Third side: \( \frac{1}{3} \times 15 = 5 \) inches
The scale copy has side lengths 3 inches, 4 inches, and 5 inches.
Identifying Scale Copies
Not every pair of figures that look similar are actually scale copies of each other. To determine whether one figure is a scale copy of another, we must check if all corresponding lengths are related by the same scale factor.
Steps to Identify Scale Copies
- Identify corresponding sides (sides that are in the same position in each figure)
- Calculate the ratio of each pair of corresponding sides
- Check if all the ratios are equal
- If all ratios are the same, the figures are scale copies and that ratio is the scale factor
Example: Rectangle A has dimensions 3 cm by 5 cm.
Rectangle B has dimensions 6 cm by 10 cm.
Rectangle C has dimensions 6 cm by 9 cm.Which rectangles are scale copies of Rectangle A?
Solution:
Check Rectangle B:
Ratio of lengths: \( \frac{6}{3} = 2 \)
Ratio of widths: \( \frac{10}{5} = 2 \)
Both ratios equal 2, so Rectangle B is a scale copy of Rectangle A with scale factor 2.
Check Rectangle C:
Ratio of lengths: \( \frac{6}{3} = 2 \)
Ratio of widths: \( \frac{9}{5} = 1.8 \)
The ratios are not equal (2 ≠ 1.8), so Rectangle C is not a scale copy of Rectangle A.
Only Rectangle B is a scale copy of Rectangle A.
Finding the Scale Factor
When we know the measurements of an original figure and its scale copy, we can find the scale factor by dividing any length in the scale copy by the corresponding length in the original figure.
\[ \text{Scale factor} = \frac{\text{Length in scale copy}}{\text{Corresponding length in original}} \]This formula works for any pair of corresponding lengths: sides, heights, diagonals, or perimeters.
Example: A photograph is 4 inches wide.
An enlarged copy of the photograph is 10 inches wide.What is the scale factor?
Solution:
Use the formula for scale factor:
Scale factor = \( \frac{\text{New width}}{\text{Original width}} = \frac{10}{4} = \frac{5}{2} = 2.5 \)
The scale factor is 2.5 or 5/2.
Scale Factor and Perimeter
An important property of scale copies is that the perimeter of the scale copy is also multiplied by the scale factor. This makes sense because the perimeter is the sum of all the side lengths, and each side length has been multiplied by the scale factor.
\[ \text{Perimeter of scale copy} = \text{Scale factor} \times \text{Perimeter of original} \]Example: A triangle has sides of 5 cm, 7 cm, and 8 cm.
You create a scale copy with a scale factor of 3.What is the perimeter of the scale copy?
Solution:
Find the perimeter of the original triangle:
Perimeter = 5 + 7 + 8 = 20 cmMultiply the original perimeter by the scale factor:
New perimeter = 3 × 20 = 60 cmThe perimeter of the scale copy is 60 cm.
Alternatively, you could find the new side lengths (15 cm, 21 cm, and 24 cm) and add them: 15 + 21 + 24 = 60 cm.
Scale Factor and Area
The relationship between scale factor and area is different from the relationship with length. When we create a scale copy, the area is multiplied by the square of the scale factor, not just the scale factor itself.
\[ \text{Area of scale copy} = (\text{Scale factor})^2 \times \text{Area of original} \]This happens because area is two-dimensional-both the length and width are being scaled, so the effect is multiplied.
Think of a square with side length 2 cm. Its area is 4 cm². If you double each side (scale factor of 2), the new square has sides of 4 cm and area of 16 cm². Notice that 16 = 4 × 4 = 4 × 2². The area was multiplied by 2² = 4, not just by 2.
Example: A rectangle has dimensions 6 m by 8 m, giving it an area of 48 m².
A scale copy is made with a scale factor of 1/2.What is the area of the scale copy?
Solution:
Square the scale factor:
\( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)Multiply the original area by the squared scale factor:
New area = \( \frac{1}{4} \times 48 = 12 \) m²The area of the scale copy is 12 m².
You can verify this: the new dimensions are 3 m by 4 m, and 3 × 4 = 12 m².
Creating Scale Copies on a Grid
When figures are drawn on a coordinate grid, we can create scale copies by multiplying the coordinates of each vertex by the scale factor. This works when the center of scaling is at the origin (0, 0).
For a point with coordinates \( (x, y) \) and a scale factor of \( k \):
\[ \text{New coordinates} = (k \times x, k \times y) \]Example: A triangle has vertices at A(2, 3), B(4, 1), and C(2, 1).
Create a scale copy with scale factor 2 centered at the origin.What are the coordinates of the new vertices?
Solution:
Multiply each coordinate by the scale factor 2:
Point A: (2 × 2, 2 × 3) = (4, 6) → A'
Point B: (2 × 4, 2 × 1) = (8, 2) → B'
Point C: (2 × 2, 2 × 1) = (4, 2) → C'
The new triangle has vertices at A'(4, 6), B'(8, 2), and C'(4, 2).
Applications of Scale Copies
Scale copies appear in many real-world situations. Understanding how they work helps us interpret and create scaled representations of objects.
Maps
Maps are scale copies of real geographic areas. A map scale tells us the relationship between distances on the map and actual distances. For example, a scale of "1 inch = 5 miles" means that every inch on the map represents 5 miles in reality.
Example: On a map with scale 1 cm = 20 km, two cities are 7.5 cm apart.
What is the actual distance between the cities?
Solution:
The scale tells us that 1 cm on the map equals 20 km in reality.
Multiply the map distance by 20:
Actual distance = 7.5 × 20 = 150 kmThe actual distance between the cities is 150 km.
Blueprints and Scale Drawings
Architects and engineers create blueprints that are scale copies of buildings or machines. These scaled-down drawings allow them to plan and communicate designs clearly.
Example: An architect's blueprint uses a scale of 1/4 inch = 1 foot.
A room on the blueprint measures 3 inches by 2.5 inches.What are the actual dimensions of the room?
Solution:
The scale means that 1/4 inch represents 1 foot, so 1 inch represents 4 feet.
Multiply each blueprint dimension by 4:
Length: 3 × 4 = 12 feet
Width: 2.5 × 4 = 10 feet
The actual room dimensions are 12 feet by 10 feet.
Models
Scale models of cars, airplanes, buildings, and other objects are scale copies. A model labeled "1:50" means that the model is 1/50 the size of the actual object, so the scale factor is 1/50.
Common Misconceptions About Scale Copies
Students sometimes make mistakes when working with scale copies. Being aware of these common errors can help you avoid them.
Misconception 1: Adding Instead of Multiplying
Some students think scaling means adding the same amount to each dimension. This is incorrect. Scaling always involves multiplying by the scale factor, not adding.
If a 3 cm by 4 cm rectangle is scaled by a factor of 2, the result is 6 cm by 8 cm (multiplying each dimension by 2), not 5 cm by 6 cm (adding 2 to each dimension).
Misconception 2: Using Scale Factor Directly for Area
Remember that area is multiplied by the square of the scale factor, not the scale factor itself. If the scale factor is 3, the area is multiplied by 9, not 3.
Misconception 3: Different Scale Factors for Different Sides
For a figure to be a true scale copy, the same scale factor must apply to all corresponding lengths. If you stretch only the width but not the height, you don't have a scale copy-you have a distorted figure.
Working Backwards: Finding Original Dimensions
Sometimes we know the dimensions of a scale copy and the scale factor, and we need to find the original dimensions. In this case, we divide by the scale factor instead of multiplying.
\[ \text{Original length} = \frac{\text{Length in scale copy}}{\text{Scale factor}} \]Example: A scale drawing of a garden has dimensions 15 inches by 12 inches.
The scale factor used was 3.What are the dimensions of the actual garden?
Solution:
Divide each dimension of the scale copy by the scale factor:
Original length = 15 ÷ 3 = 5 inches
Original width = 12 ÷ 3 = 4 inches
The original garden dimensions in the plan were 5 inches by 4 inches.
Note: If this represents a real garden, there would be an additional scale to convert these measurements to actual feet or meters.
Summary of Key Relationships
The table below summarizes how different measurements relate to the scale factor in scale copies:
Where \( k \) represents the scale factor.
Understanding scale copies helps us make sense of representations of objects at different sizes, from tiny models to large posters, from maps to blueprints. The key principle is always the same: multiply every length by the same scale factor to preserve the shape while changing the size.
