The scale factor, denoted by k, is a number that indicates how much a geometric figure is enlarged or reduced while keeping its shape unchanged. It is defined as the ratio of a corresponding length in the new figure to the corresponding length in the original figure.
Formula: Scale Factor( k ) = Dimensions of New Shape/Dimension of Original Shape
Where, k is the scale factor.
Effect of Scale Factor (k)
- If k > 1, the figure undergoes enlargement.
- If 0 < k < 1, the figure undergoes reduction.
- If k = 1, there is no change in the size of the figure.
Example: A rectangle has a length of 5 units and a width of 2 units. If the scale factor is 2, the new dimensions become 10 units and 4 units.
The figure shows how the scale factor changes the size of a rectangle:
- The original rectangle has dimensions 3 units × 2 units
- With a scale factor of 3, the dimensions become 9 units × 6 units
- With a scale factor of 1/2, the dimensions become 1.5 units × 1 unit
Types of Scale Factor
There are two main types of Scale factor:
1. Scale Up
Scaling up means increasing the size of a figure while keeping its shape unchanged. In this case, each dimension of the original figure is multiplied by a scale factor greater than 1.
Formula:
Enlargement Scale Factor = Dimensions of Enlarged Shape/Dimension of Original Shape
Example: If the original dimension is 5 units and the enlarged dimension is 15 units, then
k = 15 / 5 = 3.
This shows that each dimension is multiplied by 3, so the figure is enlarged by a scale factor of 3.
2. Scale Down
Scaling down means reducing the size of a figure while keeping its shape unchanged. In this case, each dimension of the original figure is multiplied by a scale factor between 0 and 1.
Formula:
Reduction Scale Factor = Dimensions of Reduced Shape/Dimension of Original Shape
Example: If the original dimension is 24 units and the reduced dimension is 8 units, then
k = 8 / 24 = 1/3.
This shows that each dimension is multiplied by 1/3, resulting in a smaller figure.
In the above figure, the triangles illustrate the concepts of scaled-up and scaled-down figures.
Scale Factor of a Triangle
Similar triangles have the same shape and equal corresponding angles, while their side lengths are proportional. The ratio of corresponding sides of two similar triangles is called the scale factor (k).
Key Points
- Proportional Sides: The ratio of corresponding sides is constant and gives the scale factor (k).
- Equal Angles: All corresponding angles are equal, so the shape remains unchanged.
Scale Factor of a Circle
The scale factor between two circles is determined by comparing their radii. Since all circles are similar, their size difference depends only on the ratio of their radii.
Formula:
Scale Factor = Radius of New Circle / Radius of Original Circle
Example: Consider two circles with radii 3 cm and 6 cm.
Scale Factor = 3/6 = 1/2, meaning the smaller circle is half the size of the larger circle.
Real-Life Applications
The scale factor is widely used to resize objects and maintain accurate proportions in practical situations.
- Used in maps and drawings to represent large objects in smaller sizes.
- Helps in adjusting quantities in cooking and recipes.
- Applied in comparing similar figures using corresponding sides.
- Useful for calculating percentage increase or decrease.
- Important in design and construction for accurate measurements.
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Solved Examples
Example 1: A rectangle undergoes a reduction with a scale factor of 0.5, resulting in new dimensions of 4 meters by 6 meters. Determine the dimensions of the original rectangle before the reduction.
Solution:
Given that the rectangle underwent a reduction with a scale factor of 0.5, and the new dimensions are 4 meters by 6 meters, we can find the original dimensions using the formula:
Original Dimension = New Dimension / Scale Factor
Original Length = 4m/0.5
⇒ Original Length = 8 meters
Original Width = 6m/0.5
⇒ Original Width = 12 meters
∴ The dimensions of the original rectangle before the reduction were 8 meters by 12 meters.
Example 2: A map has a scale factor of 1 inch to 5 miles. If two cities are 30 miles apart, what is the distance between them on the map?
Solution:
Given the scale factor of 1 inch to 5 miles, we can set up a proportion to find the distance on the map.
Let x be the distance on the map (in inches).
1 inch/5 miles = x/30 miles
Cross-multiplying:
⇒ 5x = 30
Dividing both sides by 5:
x = 6
So, the distance between the two cities on the map is 6 inches.
Practice Questions
Q1. A rectangle undergoes an enlargement with a scale factor of 3. If the original length is 4 meters, what is the length of the enlarged rectangle?
Q2. Two similar triangles have a scale factor of 1.5. If the shorter side of the smaller triangle is 8 centimeters, find the length of the corresponding side in the larger triangle.
Q3. A square is reduced by a scale factor of 0.7. If the original side length is 12 units, what is the length of the side in the smaller square?
Q4. An architect creates a model building with a scale factor of 1:50. If the actual building is 100 meters tall, what is the height of the model?
Q5. A photograph is enlarged with a scale factor of 2.5. If the original height is 8 inches, what is the height of the enlarged photograph?