To find the center of dilation of a square, you can employ either a graphical method by drawing lines between corresponding vertices or an analytical method using coordinate geometry and a specific formula. The center of dilation is the fixed point from which all points on a figure are scaled, either enlarged or shrunk, to create a new, similar figure.
The center of dilation acts as the vanishing point for all rays drawn from the pre-image to the image. All segments connecting a point on the original square (pre-image) to its corresponding point on the dilated square (image) will intersect at this single point.
Understanding Dilation and the Scale Factor
Before finding the center, it's crucial to understand dilation and the scale factor. Dilation is a transformation that changes the size of a figure without changing its shape. The scale factor (k) determines how much the figure is enlarged or reduced.
- If k > 1, the image is an enlargement.
- If 0 < k < 1, the image is a reduction.
- If k = 1, the image is congruent to the pre-image.
You can calculate the scale factor (k) by taking the ratio of the length of a side of the dilated square to the length of the corresponding side of the original square:
k = (Length of a side on the dilated square) / (Length of the corresponding side on the original square)
Methods to Find the Center of Dilation
There are two primary methods to determine the center of dilation for a square.
1. Graphical Method
This method is intuitive and can be performed using a ruler and pencil.
Steps:
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Identify Corresponding Vertices: Locate at least two pairs of corresponding vertices between the original square (pre-image) and the dilated square (image). For instance, if the original square has vertices A, B, C, D, and the dilated square has A', B', C', D', then A corresponds to A', B to B', and so on.
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Draw Connecting Lines: Draw a straight line segment connecting each pre-image vertex to its corresponding image vertex. For example, draw a line from A to A', another from B to B', and so forth.
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Locate the Intersection Point: The point where these lines intersect is the center of dilation. All such lines drawn from corresponding points will converge at this single center.
- Tip: For accuracy, using at least two pairs of vertices will help confirm the intersection point.
2. Analytical Method (Using Coordinates and Formula)
This method uses coordinate geometry and is precise, especially when dealing with exact coordinates.
Steps:
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Determine the Scale Factor (k):
- Choose a side from the original square and its corresponding side from the dilated square.
- Calculate the length of both sides using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)if coordinates are given. - Calculate k = (Image Side Length) / (Pre-image Side Length).
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Select Corresponding Points:
- Choose any single vertex from the original square and label its coordinates as
(x₁, y₁). - Identify its corresponding vertex on the dilated square and label its coordinates as
(x₂, y₂).
- Choose any single vertex from the original square and label its coordinates as
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Apply the Center of Dilation Formula:
- Use the scale factor k and the chosen corresponding points
(x₁, y₁)and(x₂, y₂)in the following formulas to find the coordinates of the center of dilation(x_o, y_o):
- Use the scale factor k and the chosen corresponding points
| Coordinate of Center of Dilation | Formula |
|---|---|
x-coordinate (x_o) |
x_o = (k * x₁ - x₂) / (k - 1) |
y-coordinate (y_o) |
y_o = (k * y₁ - y₂) / (k - 1) |
* These formulas allow you to directly compute the center of dilation given the scale factor and one pair of corresponding points.Example:
Let's say we have:
- Original square vertex A
(1, 1) - Dilated square vertex A'
(3, 3)(corresponding to A) - The scale factor
k = 2(meaning the dilated square's sides are twice as long as the original).
Using the formula:
x₁ = 1,y₁ = 1x₂ = 3,y₂ = 3k = 2
Calculate x_o:
x_o = (2 * 1 - 3) / (2 - 1)
x_o = (2 - 3) / 1
x_o = -1
Calculate y_o:
y_o = (2 * 1 - 3) / (2 - 1)
y_o = (2 - 3) / 1
y_o = -1
Therefore, the center of dilation is (-1, -1).
By understanding both the graphical and analytical approaches, you can accurately find the center of dilation for any square, or indeed any geometric figure, given its pre-image and image.
