Applying the Dilation Transformation

Oct 13, 2025 by windailys.com 132 views

Dilation of a Line Segment: Finding Coordinates of $A^{\prime}$ and $B^{\prime}$

A dilation is a transformation that changes the size of a figure, but not its shape. In this article, we will explore how to find the coordinates of points $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{1}{2}$ , centered at the origin, of a line segment $AB$ with endpoints $A(-6, 4)$ and $B(-8, 2)$ .

A dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation that can be represented by a scale factor. The scale factor is a ratio that describes how much the figure is enlarged or reduced. In this case, the scale factor is $\frac{1}{2}$ , which means that the figure will be reduced to half its original size.

Finding the Coordinates of $A^{\prime}$ and $B^{\prime}$

To find the coordinates of $A^{\prime}$ and $B^{\prime}$ , we need to apply the dilation transformation to the coordinates of $A$ and $B$ . The dilation transformation can be represented by the following formula:

\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}

where $(x, y)$ are the coordinates of the original point, $(x^{\prime}, y^{\prime})$ are the coordinates of the dilated point, and $k$ is the scale factor.

Applying the Dilation Transformation

We can now apply the dilation transformation to the coordinates of $A$ and $B$ .

For point $A(-6, 4)$ , we have:

\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(-6) \\ \frac{1}{2}(4) \end{pmatrix} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}

So, the coordinates of $A^{\prime}$ are $(-3, 2)$ .

For point $B(-8, 2)$ , we have:

\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(-8) \\ \frac{1}{2}(2) \end{pmatrix} = \begin{pmatrix} -4 \\ 1 \end{pmatrix}

So, the coordinates of $B^{\prime}$ are $(-4, 1)$ .

In this article, we have explored how to find the coordinates of points $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{1}{2}$ , centered at the origin, of a line segment $AB$ with endpoints $A(-6, 4)$ and $B(-8, 2)$ . We have applied the dilation transformation to the coordinates of $A$ and $B$ and found the coordinates of $A^{\prime}$ and $B^{\prime}$ to be $(-3, 2)$ and $(-4, 1)$ , respectively.

Find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{1}{3}$ , centered at the origin, of a line segment $AB$ with endpoints $A(2, 5)$ and $B(4, 3)$ .
Find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{2}{3}$ , centered at the origin, of a line segment $AB$ with endpoints $A(-2, 1)$ and $B(1, -3)$ .

For the first example problem, we have:
$\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{1}{3}(2) \\ \frac{1}{3}(5) \end{pmatrix} = \begin{pmatrix} \frac{2}{3} \\ \frac{5}{3} \end{pmatrix}$
So, the coordinates of $A^{\prime}$ are $(\frac{2}{3}, \frac{5}{3})$ .
$\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{1}{3}(4) \\ \frac{1}{3}(3) \end{pmatrix} = \begin{pmatrix} \frac{4}{3} \\ 1 \end{pmatrix}$
So, the coordinates of $B^{\prime}$ are $(\frac{4}{3}, 1)$ .
For the second example problem, we have:
$\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{2}{3}(-2) \\ \frac{2}{3}(1) \end{pmatrix} = \begin{pmatrix} -\frac{4}{3} \\ \frac{2}{3} \end{pmatrix}$
So, the coordinates of $A^{\prime}$ are $(-\frac{4}{3}, \frac{2}{3})$ .
$\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \frac{2}{3}(1) \\ \frac{2}{3}(-3) \end{pmatrix} = \begin{pmatrix} \frac{2}{3} \\ -2 \end{pmatrix}$
So, the coordinates of $B^{\prime}$ are $(\frac{2}{3}, -2)$ .

A dilation is a transformation that changes the size of a figure, but not its shape.
The dilation transformation can be represented by the formula $\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}$ , where $(x, y)$ are the coordinates of the original point, $(x^{\prime}, y^{\prime})$ are the coordinates of the dilated point, and $k$ is the scale factor.
To find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation, we need to apply the dilation transformation to the coordinates of $A$ and $B$ .

In conclusion, dilation is a transformation that changes the size of a figure, but not its shape. We have explored how to find the coordinates of points $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{1}{2}$ , centered at the origin, of a line segment $AB$ with endpoints $A(-6, 4)$ and $B(-8, 2)$ . We have applied the dilation transformation to the coordinates of $A$ and $B$ and found the coordinates of $A^{\prime}$ and $B^{\prime}$ to be $(-3, 2)$ and $(-4, 1)$ , respectively.
Dilation of a Line Segment: Q&A

In our previous article, we explored how to find the coordinates of points $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $\frac{1}{2}$ , centered at the origin, of a line segment $AB$ with endpoints $A(-6, 4)$ and $B(-8, 2)$ . In this article, we will answer some frequently asked questions about dilation of a line segment.

Q: What is 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 can be represented by a scale factor.

Q: What is a scale factor?

A: A scale factor is a ratio that describes how much the figure is enlarged or reduced. In the case of dilation, the scale factor is used to multiply the coordinates of the original point to get the coordinates of the dilated point.

Q: How do I find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation?

A: To find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation, you need to apply the dilation transformation to the coordinates of $A$ and $B$ . The dilation transformation can be represented by the formula $\begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}$ , where $(x, y)$ are the coordinates of the original point, $(x^{\prime}, y^{\prime})$ are the coordinates of the dilated point, and $k$ is the scale factor.

Q: What is the effect of a dilation with a scale factor of $\frac{1}{2}$ ?

A: A dilation with a scale factor of $\frac{1}{2}$ will reduce the size of the figure by half. This means that the coordinates of the dilated point will be half the coordinates of the original point.

Q: Can I dilate a figure by more than one scale factor?

A: Yes, you can dilate a figure by more than one scale factor. For example, if you want to dilate a figure by a scale factor of $\frac{1}{3}$ and then by a scale factor of $\frac{1}{2}$ , you can multiply the coordinates of the original point by $\frac{1}{3}$ and then by $\frac{1}{2}$ .

Q: How do I know if a dilation is centered at the origin or not?

A: If the dilation is centered at the origin, the coordinates of the dilated point will be the same as the coordinates of the original point multiplied by the scale factor. If the dilation is not centered at the origin, you will need to translate the figure to the origin before applying the dilation transformation.

Q: Can I dilate a figure by a negative scale factor?

A: Yes, you can dilate a figure by a negative scale factor. However, this will flip the figure over the origin. For example, if you dilate a figure by a scale factor of $-2$ , the coordinates of the dilated point will be the negative of the coordinates of the original point multiplied by 2.

Q: How do I find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $-2$ ?

A: To find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation with a scale factor of $-2$ , you need to multiply the coordinates of $A$ and $B$ by $-2$ . For example, if $A(-6, 4)$ and $B(-8, 2)$ , the coordinates of $A^{\prime}$ and $B^{\prime}$ will be $(-12, -8)$ and $(-16, -4)$ , respectively.

In conclusion, dilation is a transformation that changes the size of a figure, but not its shape. We have answered some frequently asked questions about dilation of a line segment and provided examples of how to find the coordinates of $A^{\prime}$ and $B^{\prime}$ after a dilation.