Given $\triangle ABC$ With Coordinates $A(-5, -7)$, $B(6, -3)$, And $C(2, 7)$, Find The Coordinates Of Its Image After A Dilation Centered At The Origin With A Scale Factor Of 2.A. $A(-5, -7), B(6,
Introduction
In geometry, a dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation that involves scaling the figure by a certain factor. In this article, we will explore the concept of dilation and how to find the coordinates of the image of a triangle after a dilation centered at the origin with a scale factor of 2.
What is a Dilation?
A dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation that involves scaling the figure by a certain factor. In other words, a dilation is a transformation that stretches or shrinks a figure by a certain scale factor. The scale factor is a number that determines how much the figure is stretched or shrunk.
Dilation Centered at the Origin
A dilation centered at the origin is a transformation that involves scaling the figure by a certain factor with respect to the origin. The origin is the point (0, 0) on the coordinate plane. When a figure is dilated centered at the origin, the distance between each point of the figure and the origin is multiplied by the scale factor.
Scale Factor of 2
In this article, we will be dealing with a scale factor of 2. This means that the distance between each point of the triangle and the origin will be multiplied by 2. As a result, the coordinates of the image of the triangle will be twice the coordinates of the original triangle.
Finding the Coordinates of the Image
To find the coordinates of the image of the triangle after a dilation centered at the origin with a scale factor of 2, we need to multiply the coordinates of each point of the triangle by 2.
Step 1: Multiply the x-coordinate of each point by 2
The x-coordinate of each point is multiplied by 2 to get the new x-coordinate of the image.
Step 2: Multiply the y-coordinate of each point by 2
The y-coordinate of each point is multiplied by 2 to get the new y-coordinate of the image.
Step 3: Write the new coordinates of each point
The new coordinates of each point are written as (2x, 2y), where x and y are the original coordinates of the point.
Example: Dilation of Triangle ABC
Let's consider the triangle ABC with coordinates A(-5, -7), B(6, -3), and C(2, 7). We want to find the coordinates of its image after a dilation centered at the origin with a scale factor of 2.
Step 1: Multiply the x-coordinate of each point by 2
The x-coordinate of each point is multiplied by 2 to get the new x-coordinate of the image.
| Point | Original x-coordinate | New x-coordinate |
|---|---|---|
| A | -5 | -10 |
| B | 6 | 12 |
| C | 2 | 4 |
Step 2: Multiply the y-coordinate of each point by 2
The y-coordinate of each point is multiplied by 2 to get the new y-coordinate of the image.
| Point | Original y-coordinate | New y-coordinate |
|---|---|---|
| A | -7 | -14 |
| B | -3 | -6 |
| C | 7 | 14 |
Step 3: Write the new coordinates of each point
The new coordinates of each point are written as (2x, 2y), where x and y are the original coordinates of the point.
| Point | Original coordinates | New coordinates |
|---|---|---|
| A | (-5, -7) | (-10, -14) |
| B | (6, -3) | (12, -6) |
| C | (2, 7) | (4, 14) |
Conclusion
In this article, we have explored the concept of dilation and how to find the coordinates of the image of a triangle after a dilation centered at the origin with a scale factor of 2. We have seen that the coordinates of the image are obtained by multiplying the coordinates of each point of the triangle by the scale factor. We have also seen that the distance between each point of the triangle and the origin is multiplied by the scale factor.
Introduction
In our previous article, we explored the concept of dilation and how to find the coordinates of the image of a triangle after a dilation centered at the origin with a scale factor of 2. In this article, we will answer some frequently asked questions about dilation and provide additional examples to help you understand the concept better.
Q&A
Q1: What is the difference between a dilation and a translation?
A dilation is a transformation that changes the size of a figure, but not its shape. A translation, on the other hand, is a transformation that moves a figure from one location to another without changing its size or shape.
Q2: How do I determine the scale factor of a dilation?
The scale factor of a dilation is the number that determines how much the figure is stretched or shrunk. It is usually given as a ratio or a decimal value.
Q3: Can a dilation be centered at a point other than the origin?
Yes, a dilation can be centered at any point on the coordinate plane. However, the distance between each point of the figure and the center of dilation is multiplied by the scale factor.
Q4: How do I find the coordinates of the image of a triangle after a dilation?
To find the coordinates of the image of a triangle after a dilation, you need to multiply the coordinates of each point of the triangle by the scale factor.
Q5: Can a dilation be combined with other transformations?
Yes, a dilation can be combined with other transformations such as translations, rotations, and reflections. However, the order in which the transformations are applied is important.
Q6: How do I determine the type of dilation (enlargement or reduction) based on the scale factor?
If the scale factor is greater than 1, the dilation is an enlargement. If the scale factor is less than 1, the dilation is a reduction.
Q7: Can a dilation be represented graphically?
Yes, a dilation can be represented graphically by drawing the original figure and the image figure on the same coordinate plane.
Q8: How do I find the coordinates of the image of a point after a dilation?
To find the coordinates of the image of a point after a dilation, you need to multiply the coordinates of the point by the scale factor.
Q9: Can a dilation be used to solve real-world problems?
Yes, a dilation can be used to solve real-world problems such as designing buildings, bridges, and other structures.
Q10: How do I determine the scale factor of a dilation in a real-world problem?
The scale factor of a dilation in a real-world problem is usually given as a ratio or a decimal value. You need to use this value to determine the size of the image figure.
Examples
Example 1: Dilation of a Triangle
Let's consider the triangle ABC with coordinates A(-5, -7), B(6, -3), and C(2, 7). We want to find the coordinates of its image after a dilation centered at the origin with a scale factor of 3.
| Point | Original coordinates | New coordinates |
|---|---|---|
| A | (-5, -7) | (-15, -21) |
| B | (6, -3) | (18, -9) |
| C | (2, 7) | (6, 21) |
Example 2: Dilation of a Point
Let's consider the point P(3, 4). We want to find the coordinates of its image after a dilation centered at the origin with a scale factor of 2.
| Point | Original coordinates | New coordinates |
|---|---|---|
| P | (3, 4) | (6, 8) |
Conclusion
In this article, we have answered some frequently asked questions about dilation and provided additional examples to help you understand the concept better. We have seen that a dilation is a transformation that changes the size of a figure, but not its shape, and that the coordinates of the image of a triangle after a dilation can be found by multiplying the coordinates of each point of the triangle by the scale factor.