Enlargement is a mathematical concept that involves scaling an object or shape by a certain factor. It is a transformation that changes the size of an object while maintaining its shape and proportions. Enlargement is commonly used in various fields such as geometry, physics, and engineering to represent real-world scenarios or to simplify calculations.
Enlargement involves several key knowledge points, including:
Scale factor: The scale factor is the ratio of the size of the enlarged shape to the original shape. It determines how much the shape is scaled up or down. The scale factor can be greater than 1 for enlargement or less than 1 for reduction.
Center of enlargement: The center of enlargement is the point from which the shape is scaled. All points on the shape are enlarged or reduced by the same scale factor from this center.
Proportional relationships: Enlargement maintains the proportional relationships between the corresponding sides of the original and enlarged shapes. This means that the ratios of corresponding side lengths remain the same after the enlargement.
The process of enlargement can be explained step by step as follows:
Identify the scale factor: Determine whether the shape is being enlarged or reduced and find the scale factor. The scale factor can be given directly or calculated by comparing the lengths of corresponding sides.
Locate the center of enlargement: Identify the point from which the shape is being scaled. This point is usually given or can be determined based on the problem context.
Determine the new coordinates: Multiply the coordinates of each point on the shape by the scale factor, considering the center of enlargement as the origin. This will give the new coordinates of the enlarged shape.
Plot the new shape: Use the new coordinates to plot the enlarged shape on a graph or draw it accurately if working on paper.
The formula for enlargement involves multiplying the coordinates of each point by the scale factor, considering the center of enlargement as the origin. If the original coordinates of a point are (x, y) and the scale factor is k, the new coordinates after enlargement can be calculated using the following formula:
New x-coordinate = k * (x - center of enlargement x-coordinate) New y-coordinate = k * (y - center of enlargement y-coordinate)
To apply the enlargement formula, follow these steps:
The symbol for enlargement is usually a capital letter "E" with a subscript indicating the scale factor. For example, E2 represents an enlargement with a scale factor of 2.
There are several methods for performing an enlargement:
Using a grid: Draw a grid on the original shape and enlarge it by multiplying the coordinates by the scale factor. This method is useful for visualizing the transformation and maintaining accuracy.
Using a ruler and compass: Use a ruler and compass to construct the enlarged shape based on the given scale factor and center of enlargement. This method is more precise and can be used when working on paper.
Using coordinate geometry: Apply the enlargement formula to calculate the new coordinates of each point and plot the enlarged shape on a graph. This method is suitable for working with coordinates and equations.
Example 1: Given a triangle ABC with coordinates A(2, 3), B(4, 1), and C(6, 5). Enlarge the triangle with a scale factor of 2 and center of enlargement at point (3, 2).
Solution:
Calculate the new coordinates for point A: New x-coordinate = 2 * (2 - 3) = -2 New y-coordinate = 2 * (3 - 2) = 2 New coordinates for A: (-2, 2)
Calculate the new coordinates for point B: New x-coordinate = 2 * (4 - 3) = 2 New y-coordinate = 2 * (1 - 2) = -2 New coordinates for B: (2, -2)
Calculate the new coordinates for point C: New x-coordinate = 2 * (6 - 3) = 6 New y-coordinate = 2 * (5 - 2) = 6 New coordinates for C: (6, 6)
The enlarged triangle has vertices at (-2, 2), (2, -2), and (6, 6).
Example 2: A rectangle has vertices at A(1, 2), B(5, 2), C(5, 6), and D(1, 6). Enlarge the rectangle with a scale factor of 3 and center of enlargement at the origin.
Solution:
Calculate the new coordinates for point A: New x-coordinate = 3 * (1 - 0) = 3 New y-coordinate = 3 * (2 - 0) = 6 New coordinates for A: (3, 6)
Calculate the new coordinates for point B: New x-coordinate = 3 * (5 - 0) = 15 New y-coordinate = 3 * (2 - 0) = 6 New coordinates for B: (15, 6)
Calculate the new coordinates for point C: New x-coordinate = 3 * (5 - 0) = 15 New y-coordinate = 3 * (6 - 0) = 18 New coordinates for C: (15, 18)
Calculate the new coordinates for point D: New x-coordinate = 3 * (1 - 0) = 3 New y-coordinate = 3 * (6 - 0) = 18 New coordinates for D: (3, 18)
The enlarged rectangle has vertices at (3, 6), (15, 6), (15, 18), and (3, 18).
Enlarge a square with side length 4 units by a scale factor of 2 and center of enlargement at point (2, 2).
A triangle has vertices at A(1, 1), B(3, 1), and C(2, 4). Enlarge the triangle with a scale factor of 0.5 and center of enlargement at the origin.
A circle with center at (2, 3) and radius 5 units is enlarged by a scale factor of 3. Find the new center and radius of the enlarged circle.
Question: What is the difference between enlargement and reduction?
Answer: Enlargement and reduction are both transformations that change the size of an object or shape. The main difference is that enlargement increases the size of the shape, while reduction decreases it. Enlargement involves multiplying the coordinates by a scale factor greater than 1, while reduction involves multiplying by a scale factor less than 1. Both transformations maintain the shape and proportions of the original object.