What if you were given the coordinates of a figure and were asked to dilate that figure by a scale factor of 2? How could you find the coordinates of the dilated figure? After completing this Concept, you'll be able to solve problems like this one.

### Watch This

CK-12 Foundation: Chapter7DilationintheCoordinatePlaneA

### Guidance

A **dilation** makes a figure larger or smaller, but has the same shape as the original. In other words, the dilation is similar to the original. All dilations have a **center** and a **scale factor.** The center is the point of reference for the dilation (like the vanishing point in a perspective drawing) and scale factor tells us how much the figure stretches or shrinks. A scale factor is typically labeled \begin{align*}k\end{align*} and is always greater than zero. Also, if the original figure is labeled \begin{align*}\triangle ABC\end{align*}, for example, the dilation would be \begin{align*}\triangle A'B'C'\end{align*}. The ‘ indicates that it is a copy. This tic mark is said “prime,” so \begin{align*}A'\end{align*} is read “A prime.” A second dilation would be \begin{align*}A''\end{align*}, read “A double-prime.”

*If the dilated image is smaller than the original, then the scale factor is \begin{align*}0 < k < 1\end{align*}.*

*If the dilated image is larger than the original, then the scale factor is \begin{align*}k>1\end{align*}.*

To dilate something in the coordinate plane, multiply each coordinate by the scale factor. This is called ** mapping.** For any dilation the mapping will be \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}. In this Concept, the center of dilation will always be the origin, unless otherwise stated.

#### Example A

Determine the coordinates of \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle A'B'C'\end{align*} and find the scale factor.

The coordinates of \begin{align*}\triangle ABC\end{align*} are \begin{align*}A(2, 1), B(5, 1)\end{align*} and \begin{align*}C(3, 6)\end{align*}. The coordinates of \begin{align*}\triangle A'B'C'\end{align*} are \begin{align*}A'(6, 3), B'(15, 3)\end{align*} and \begin{align*}C'(9, 18)\end{align*}. By looking at the corresponding coordinates, each is three times the original. That means \begin{align*}k = 3\end{align*}.

Again, the center, original point, and dilated point are collinear. Therefore, you can draw a ray from the origin to \begin{align*}C', B',\end{align*} and \begin{align*}A'\end{align*} such that the rays pass through \begin{align*}C, B,\end{align*} and \begin{align*}A\end{align*}, respectively.

#### Example B

Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example A.

\begin{align*}& \underline{\triangle ABC} && \underline{\triangle A'B'C'}\\ & AB = \sqrt{(2-5)^2 + (1-1)^2} = \sqrt{9} =3 && A'B' = \sqrt{(6-15)^2 + (3-3)^2} = \sqrt{81} = 9\\ & AC = \sqrt{(2-3)^2 + (1-6)^2} = \sqrt{26} && A'C' = \sqrt{(6-9)^2+(3-18)^2} = \sqrt{234} = 3 \sqrt{26}\\ & CB = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{29} && C'B' = \sqrt{(9-15)^2 +(18-3)^2} = \sqrt{261} = 3 \sqrt{29}\end{align*}

From this, we also see that all the sides of \begin{align*}\triangle A'B'C'\end{align*} are three times larger than \begin{align*}\triangle ABC\end{align*}.

#### Example C

Quadrilateral \begin{align*}EFGH\end{align*} has vertices \begin{align*}E(-4, -2), F(1, 4), G(6, 2)\end{align*} and \begin{align*}H(0, -4)\end{align*}. Draw the dilation with a scale factor of 1.5.

Remember that to dilate something in the coordinate plane, multiply each coordinate by the scale factor.

For this dilation, the mapping will be \begin{align*}(x, y) \rightarrow (1.5x, 1.5y)\end{align*}.

\begin{align*}& E(-4, -2) \rightarrow (1.5(-4), 1.5(-2)) \rightarrow E'(-6, -3)\\ & F(1, 4) \rightarrow (1.5(1), 1.5(4)) \rightarrow F'(1.5, 6)\\ & G(6, 2) \rightarrow (1.5(6), 1.5(2)) \rightarrow G'(9,3)\\ & H(0, -4) \rightarrow (1.5(0),1.5(-4)) \rightarrow H'(0, -6)\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter7DilationintheCoordinatePlaneB

### Guided Practice

Given \begin{align*}A\end{align*} and the scale factor, determine the coordinates of the dilated point, \begin{align*}A'\end{align*}. You may assume the center of dilation is the origin.

1. \begin{align*}A(3, 9), k = \frac{2}{3}\end{align*}

2. \begin{align*}A(-4, 6), k = 2\end{align*}

3. \begin{align*}A(9, -13), k = \frac{1}{2}\end{align*}

**Answers**

Remember that the mapping will be \begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}.

1. \begin{align*}A' (2, 6)\end{align*}

2. \begin{align*}A'(-8, 12)\end{align*}

3. \begin{align*}A'(4.5, -6.5)\end{align*}

### Explore More

Given \begin{align*}A\end{align*} and \begin{align*}A'\end{align*}, find the scale factor. You may assume the center of dilation is the origin.

- \begin{align*}A(8, 2), A'(12, 3)\end{align*}
- \begin{align*}A(-5, -9), A'(-45, -81)\end{align*}
- \begin{align*}A(22, -7), A'(11, -3.5)\end{align*}

The origin is the center of dilation. Find the coordinates of the dilation of each figure, given the scale factor.

- \begin{align*}A(2, 4), B(-3, 7), C(-1, -2); k = 3\end{align*}
- \begin{align*}A(12, 8), B(-4, -16), C(0, 10); k = \frac{3}{4}\end{align*}

** Multi-Step Problem** Questions 6-12 build upon each other.

- Plot \begin{align*}A(1, 2), B(12, 4), C(10, 10)\end{align*}. Connect to form a triangle.
- Make the origin the center of dilation. Draw 4 rays from the origin to each point from #6. Then, plot \begin{align*}A'(2, 4), B'(24, 8), C'(20, 20)\end{align*}. What is the scale factor?
- Use \begin{align*}k =4\end{align*}, to find \begin{align*}A''B''C''\end{align*}. Plot these points.
- What is the scale factor from \begin{align*}A'B'C'\end{align*} to \begin{align*}A''B''C''\end{align*}?
- Find (\begin{align*}O\end{align*} is the origin):
- \begin{align*}OA\end{align*}
- \begin{align*}AA'\end{align*}
- \begin{align*}AA''\end{align*}
- \begin{align*}OA'\end{align*}
- \begin{align*}OA''\end{align*}

- Find:
- \begin{align*}AB\end{align*}
- \begin{align*}A'B'\end{align*}
- \begin{align*}A''B''\end{align*}

- Compare the ratios:
- \begin{align*}OA:OA'\end{align*} and \begin{align*}AB:A'B'\end{align*}
- \begin{align*}OA:OA''\end{align*} and \begin{align*}AB:A''B''\end{align*}

For questions 13-18, use quadrilateral \begin{align*}ABCD\end{align*} with \begin{align*}A(1, 5), B(2, 6), C(3, 3)\end{align*} and \begin{align*}D(1, 3)\end{align*} and its transformation \begin{align*}A'B'C'D'\end{align*} with \begin{align*}A'(-3, 1), B'(0, 4), C'(3, -5)\end{align*} and \begin{align*}D'(-3, -5)\end{align*}.

- Plot the two quadrilaterals in the coordinate plane.
- Find the equation of \begin{align*}\overleftrightarrow{CC'}\end{align*}.
- Find the equation of \begin{align*}\overleftrightarrow{DD'}\end{align*}.
- Find the intersection of these two lines algebraically or graphically.
- What is the significance of this point?
- What is the scale factor of the dilation?