1) Dilate $\triangle IJK$ By A Scale Factor Of $\frac{1}{2}$ About The Point $(-6,6$\].$\begin{array}{l} I = (-4, 4) \quad I^{\prime} = (\square, \square) \\ J = (6, 2) \quad J^{\prime} = (\square, \square) \\ K = (2,

Dilating a Triangle: Understanding the Concept and Calculating New Coordinates

Introduction to Dilations

A dilation is a transformation that changes the size of a figure, but not its shape. In this article, we will focus on dilating a triangle by a scale factor of $\frac{1}{2}$ about a given point. This concept is crucial in mathematics, particularly in geometry and trigonometry.

What is a Dilation?

A dilation is a transformation that enlarges or reduces a figure by a scale factor. The scale factor is a number that represents how much the figure is enlarged or reduced. In this case, we are dilating the triangle $\triangle IJK$ by a scale factor of $\frac{1}{2}$, which means that the new coordinates of the vertices will be half the distance from the center of dilation.

Understanding the Center of Dilation

The center of dilation is the point about which the dilation takes place. In this case, the center of dilation is the point $(-6,6)$. This point remains fixed while the triangle is dilated.

Calculating New Coordinates

To calculate the new coordinates of the vertices of the triangle, we need to use the formula for dilation:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h \\ k \end{pmatrix} \cdot (k - 1)$$

where $(x, y)$ are the original coordinates of the vertex, $(h, k)$ are the coordinates of the center of dilation, and $k$ is the scale factor.

Calculating the New Coordinates of Vertex I

The original coordinates of vertex $I$ are $(-4, 4)$. The center of dilation is $(-6, 6)$, and the scale factor is $\frac{1}{2}$. Using the formula for dilation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -4 \\ 4 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(\frac{1}{2} - 1\right)$$

Simplifying the equation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -4 \\ 4 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(-\frac{1}{2}\right)$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -4 \\ 4 \end{pmatrix} + \begin{pmatrix} 3 \\ -3 \end{pmatrix}$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$$

Therefore, the new coordinates of vertex $I$ are $(-1, 1)$.

Calculating the New Coordinates of Vertex J

The original coordinates of vertex $J$ are $(6, 2)$. The center of dilation is $(-6, 6)$, and the scale factor is $\frac{1}{2}$. Using the formula for dilation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(\frac{1}{2} - 1\right)$$

Simplifying the equation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(-\frac{1}{2}\right)$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ -3 \end{pmatrix}$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 9 \\ -1 \end{pmatrix}$$

Therefore, the new coordinates of vertex $J$ are $(9, -1)$.

Calculating the New Coordinates of Vertex K

The original coordinates of vertex $K$ are $(2, 8)$. The center of dilation is $(-6, 6)$, and the scale factor is $\frac{1}{2}$. Using the formula for dilation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 \\ 8 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(\frac{1}{2} - 1\right)$$

Simplifying the equation, we get:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 \\ 8 \end{pmatrix} + \begin{pmatrix} -6 \\ 6 \end{pmatrix} \cdot \left(-\frac{1}{2}\right)$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 \\ 8 \end{pmatrix} + \begin{pmatrix} 3 \\ -3 \end{pmatrix}$$

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$$

Therefore, the new coordinates of vertex $K$ are $(5, 5)$.

Conclusion

In this article, we have discussed the concept of dilation and how to calculate the new coordinates of the vertices of a triangle after dilation. We have used the formula for dilation to calculate the new coordinates of the vertices of the triangle $\triangle IJK$ after dilation by a scale factor of $\frac{1}{2}$ about the point $(-6,6)$. The new coordinates of the vertices are $I'(-1,1)$, $J'(9,-1)$, and $K'(5,5)$.
Dilating a Triangle: Q&A

Q: What is a dilation?

A: A dilation is a transformation that changes the size of a figure, but not its shape. It is a type of transformation that enlarges or reduces a figure by a scale factor.

Q: What is the center of dilation?

A: The center of dilation is the point about which the dilation takes place. It is the point that remains fixed while the figure is dilated.

Q: How do I calculate the new coordinates of the vertices of a triangle after dilation?

A: To calculate the new coordinates of the vertices of a triangle after dilation, you can use the formula for dilation:

$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h \\ k \end{pmatrix} \cdot (k - 1)$$

where $(x, y)$ are the original coordinates of the vertex, $(h, k)$ are the coordinates of the center of dilation, and $k$ is the scale factor.

Q: What is the scale factor?

A: The scale factor is a number that represents how much the figure is enlarged or reduced. It is a value between 0 and 1 for a reduction and greater than 1 for an enlargement.

Q: How do I determine the scale factor?

A: The scale factor is determined by the problem. In the case of the triangle $\triangle IJK$, the scale factor is $\frac{1}{2}$, which means that the new coordinates of the vertices will be half the distance from the center of dilation.

Q: What is the difference between a dilation and a translation?

A: A dilation is a transformation that changes the size of a figure, but not its shape. A translation is a transformation that moves a figure from one position to another without changing its size or shape.

Q: Can I dilate a figure by a scale factor of 0?

A: No, you cannot dilate a figure by a scale factor of 0. A scale factor of 0 would mean that the figure is not changed at all, which is not a dilation.

Q: Can I dilate a figure by a scale factor of 1?

A: Yes, you can dilate a figure by a scale factor of 1. A scale factor of 1 means that the figure is not changed at all, which is a dilation.

Q: How do I know if a dilation is an enlargement or a reduction?

A: To determine if a dilation is an enlargement or a reduction, you need to look at the scale factor. If the scale factor is greater than 1, the dilation is an enlargement. If the scale factor is between 0 and 1, the dilation is a reduction.

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

A: No, you cannot dilate a figure by a negative scale factor. A negative scale factor would mean that the figure is reflected across the center of dilation, which is not a dilation.

Q: Can I dilate a figure by a scale factor of -1?

A: No, you cannot dilate a figure by a scale factor of -1. A scale factor of -1 would mean that the figure is reflected across the center of dilation, which is not a dilation.

Conclusion

In this Q&A article, we have discussed various questions related to dilating a triangle. We have covered topics such as the definition of a dilation, the center of dilation, calculating new coordinates, scale factors, and more. We hope that this article has been helpful in answering your questions and providing a better understanding of dilations.