What Is The Scale Factor In The Dilation If The Coordinates Of { A^{\prime} $}$ Are { (-7, 6)$}$ And The Coordinates Of { C^{\prime} $}$ Are { (-4, 3)$}$?

Mar 12, 2025 by ADMIN 155 views

What is the Scale Factor in the Dilation?

Understanding Dilation and Scale Factor

Dilation is a transformation that changes the size of a figure. It involves scaling the original figure by a certain factor, which is known as the scale factor. The scale factor is a ratio that compares the size of the original figure to the size of the transformed figure. In this article, we will explore how to find the scale factor in a dilation given the coordinates of two points.

What is a Scale Factor?

A scale factor is a ratio that compares the size of the original figure to the size of the transformed figure. It is a measure of how much the figure has been enlarged or reduced. The scale factor is always greater than or equal to 1, and it is usually represented by the letter "k".

Finding the Scale Factor in a Dilation

To find the scale factor in a dilation, we need to use the coordinates of two points on the original figure and the transformed figure. Let's consider the example given in the problem: the coordinates of point A' are (-7, 6) and the coordinates of point C' are (-4, 3).

Step 1: Identify the Coordinates of the Two Points

The coordinates of point A' are (-7, 6) and the coordinates of point C' are (-4, 3). These are the two points that we will use to find the scale factor.

Step 2: Calculate the Distance Between the Two Points

To find the scale factor, we need to calculate the distance between the two points on the original figure and the distance between the two points on the transformed figure. We can use the distance formula to calculate these distances.

Step 3: Use the Distance Formula to Calculate the Distances

The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

We can use this formula to calculate the distance between the two points on the original figure and the distance between the two points on the transformed figure.

Step 4: Calculate the Distance Between the Two Points on the Original Figure

Using the distance formula, we can calculate the distance between the two points on the original figure:

d1 = √((-4 - (-7))^2 + (3 - 6)^2) d1 = √((3)^2 + (-3)^2) d1 = √(9 + 9) d1 = √18

Step 5: Calculate the Distance Between the Two Points on the Transformed Figure

Using the distance formula, we can calculate the distance between the two points on the transformed figure:

d2 = √((-7 - (-4))^2 + (6 - 3)^2) d2 = √((-3)^2 + (3)^2) d2 = √(9 + 9) d2 = √18

Step 6: Find the Scale Factor

Now that we have calculated the distances between the two points on the original figure and the transformed figure, we can find the scale factor. The scale factor is given by the ratio of the distance between the two points on the transformed figure to the distance between the two points on the original figure.

k = d2 / d1 k = (√18) / (√18) k = 1

Conclusion

In this article, we have explored how to find the scale factor in a dilation given the coordinates of two points. We have used the distance formula to calculate the distances between the two points on the original figure and the transformed figure, and then found the scale factor by taking the ratio of these distances. The scale factor is a measure of how much the figure has been enlarged or reduced, and it is an important concept in geometry and mathematics.

Example Problems

Find the scale factor in a dilation if the coordinates of point A' are (2, 5) and the coordinates of point C' are (3, 7).
Find the scale factor in a dilation if the coordinates of point A' are (-2, 3) and the coordinates of point C' are (-1, 4).

Solutions to Example Problems

To find the scale factor in this dilation, we can use the same steps as before. We can calculate the distance between the two points on the original figure and the distance between the two points on the transformed figure, and then find the scale factor by taking the ratio of these distances.
k = d2 / d1
k = (√(3 - 2)^2 + (7 - 5)^2) / (√(2 - (-2))^2 + (3 - 3)^2)
k = (√(1)^2 + (2)^2) / (√(4)^2 + (0)^2)
k = (√1 + 4) / (√16)
k = (√5) / 4
k = 0.79 (rounded to two decimal places)
To find the scale factor in this dilation, we can use the same steps as before. We can calculate the distance between the two points on the original figure and the distance between the two points on the transformed figure, and then find the scale factor by taking the ratio of these distances.
k = d2 / d1
k = (√(-1 - (-2))^2 + (4 - 3)^2) / (√(-2 - (-2))^2 + (3 - 3)^2)
k = (√(1)^2 + (1)^2) / (√(0)^2 + (0)^2)
k = (√1 + 1) / (√0)
k = (√2) / 0
This is undefined, so we cannot find the scale factor in this dilation.

Final Thoughts

Q: What is a scale factor in dilation?

A: A scale factor is a ratio that compares the size of the original figure to the size of the transformed figure. It is a measure of how much the figure has been enlarged or reduced.

Q: How do I find the scale factor in a dilation?

A: To find the scale factor in a dilation, you need to use the coordinates of two points on the original figure and the transformed figure. You can use the distance formula to calculate the distances between the two points on the original figure and the transformed figure, and then find the scale factor by taking the ratio of these distances.

Q: What is the formula for finding the scale factor?

A: The formula for finding the scale factor is:

k = d2 / d1

where k is the scale factor, d2 is the distance between the two points on the transformed figure, and d1 is the distance between the two points on the original figure.

Q: What if the scale factor is not a whole number?

A: If the scale factor is not a whole number, it means that the figure has been enlarged or reduced by a fraction of the original size. For example, if the scale factor is 1.5, it means that the figure has been enlarged by 50% of the original size.

Q: Can I find the scale factor if I only know the coordinates of one point on the original figure and one point on the transformed figure?

A: No, you cannot find the scale factor if you only know the coordinates of one point on the original figure and one point on the transformed figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is negative?

A: If the scale factor is negative, it means that the figure has been reflected as well as enlarged or reduced. For example, if the scale factor is -2, it means that the figure has been enlarged by a factor of 2 and then reflected across the x-axis.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and one point on the original figure and one point on the transformed figure?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and one point on the original figure and one point on the transformed figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is 1?

A: If the scale factor is 1, it means that the figure has not been enlarged or reduced. The figure remains the same size as the original figure.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the original figure?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and the coordinates of one point on the original figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is 0?

A: If the scale factor is 0, it means that the figure has been reduced to a point. The figure has no size or dimension.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the transformed figure?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and the coordinates of one point on the transformed figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is undefined?

A: If the scale factor is undefined, it means that the figure has been reflected across the x-axis or y-axis, but not enlarged or reduced. The figure remains the same size as the original figure.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of two points on the original figure?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and the coordinates of two points on the original figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is infinity?

A: If the scale factor is infinity, it means that the figure has been enlarged to an infinite size. The figure has no bounds or limits.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of two points on the transformed figure?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and the coordinates of two points on the transformed figure. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is a fraction?

A: If the scale factor is a fraction, it means that the figure has been enlarged or reduced by a fraction of the original size. For example, if the scale factor is 1/2, it means that the figure has been reduced to half of the original size.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are not on the same line?

A: No, you cannot find the scale factor if you only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are not on the same line. You need to know the coordinates of two points on the original figure and the transformed figure to find the scale factor.

Q: What if the scale factor is a decimal?

A: If the scale factor is a decimal, it means that the figure has been enlarged or reduced by a decimal fraction of the original size. For example, if the scale factor is 0.5, it means that the figure has been reduced to half of the original size.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are on the same line?

A: Yes, you can find the scale factor if you only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are on the same line. You can use the formula:

k = (x2 - x1) / (x2' - x1')

where k is the scale factor, x1 and x2 are the x-coordinates of the two points on the original figure, and x1' and x2' are the x-coordinates of the two points on the transformed figure.

Q: What if the scale factor is a negative fraction?

A: If the scale factor is a negative fraction, it means that the figure has been reflected across the x-axis or y-axis, and then enlarged or reduced by a fraction of the original size. For example, if the scale factor is -1/2, it means that the figure has been reflected across the x-axis and then reduced to half of the original size.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are on the same line and the scale factor is a negative fraction?

k = (x2 - x1) / (x2' - x1')

where k is the scale factor, x1 and x2 are the x-coordinates of the two points on the original figure, and x1' and x2' are the x-coordinates of the two points on the transformed figure.

Q: What if the scale factor is a negative decimal?

A: If the scale factor is a negative decimal, it means that the figure has been reflected across the x-axis or y-axis, and then enlarged or reduced by a decimal fraction of the original size. For example, if the scale factor is -0.5, it means that the figure has been reflected across the x-axis and then reduced to half of the original size.

Q: Can I find the scale factor if I only know the coordinates of the center of dilation and the coordinates of one point on the original figure and one point on the transformed figure, but the points are on the same line and the scale factor is a negative decimal?

k = (x2 - x1) / (x2' - x1')

where k is the scale factor, x1 and x2 are the x-coordinates of the two points on the original figure, and x1' and x2' are the x-coordinates of the two points on the transformed figure.

Q: What if the scale factor is a fraction with a negative numerator and a positive denominator?

A: If