The Endpoints Of A B ‾ \overline{AB} A B Are A ( 2 , 2 A(2,2 A ( 2 , 2 ] And B ( 3 , 8 B(3,8 B ( 3 , 8 ]. A B ‾ \overline{AB} A B Is Dilated By A Scale Factor Of 3.5 With The Origin As The Center Of Dilation To Give The Image A ′ B ′ ‾ \overline{A^{\prime}B^{\prime}} A ′ B ′ .

Introduction

In geometry, dilation is a transformation that changes the size of a figure. When a figure is dilated, its size is increased or decreased by a scale factor. In this article, we will discuss the endpoints of a dilated line segment and how to find them using the scale factor and the coordinates of the original endpoints.

What is Dilation?

Dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that preserves the shape of the figure but changes its size. When a figure is dilated, its size is increased or decreased by a scale factor. The scale factor is a number that represents the ratio of the new size to the original size.

The Scale Factor

The scale factor is a number that represents the ratio of the new size to the original size. It is used to determine the size of the dilated figure. In this article, we will use a scale factor of 3.5 to dilate the line segment $\overline{AB}$.

The Original Line Segment

The original line segment $\overline{AB}$ has endpoints $A(2,2)$ and $B(3,8)$. We can plot these points on a coordinate plane to visualize the line segment.

Plotting the Original Line Segment

To plot the original line segment, we need to plot the points $A(2,2)$ and $B(3,8)$ on a coordinate plane.

import matplotlib.pyplot as plt
A = (2, 2)
B = (3, 8)

plt.scatter(A[0], A[1], color='blue')
plt.scatter(B[0], B[1], color='red')

plt.plot([A[0], B[0]], [A[1], B[1]], color='black')

plt.show()

Dilating the Line Segment

To dilate the line segment $\overline{AB}$, we need to multiply the coordinates of the endpoints by the scale factor of 3.5.

Dilating the Endpoints

To dilate the endpoints, we need to multiply their coordinates by the scale factor of 3.5.

# Define the scale factor
scale_factor = 3.5

A_prime = (A[0] * scale_factor, A[1] * scale_factor)
B_prime = (B[0] * scale_factor, B[1] * scale_factor)

print("A' =", A_prime)
print("B' =", B_prime)

The Dilated Line Segment

The dilated line segment $\overline{A^{\prime}B^{\prime}}$ has endpoints $A^{\prime}(7,7)$ and $B^{\prime}(10.5,28)$. We can plot these points on a coordinate plane to visualize the dilated line segment.

Plotting the Dilated Line Segment

To plot the dilated line segment, we need to plot the points $A^{\prime}(7,7)$ and $B^{\prime}(10.5,28)$ on a coordinate plane.

import matplotlib.pyplot as plt

A_prime = (7, 7)
B_prime = (10.5, 28)

plt.scatter(A_prime[0], A_prime[1], color='blue')
plt.scatter(B_prime[0], B_prime[1], color='red')

plt.plot([A_prime[0], B_prime[0]], [A_prime[1], B_prime[1]], color='black')

plt.show()

Conclusion

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Introduction

In our previous article, we discussed the endpoints of a dilated line segment and how to find them using the scale factor and the coordinates of the original endpoints. In this article, we will answer some frequently asked questions about dilating line segments.

Q: What is dilation?

A: Dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that preserves the shape of the figure but changes its size. When a figure is dilated, its size is increased or decreased by a scale factor.

Q: What is a scale factor?

A: The scale factor is a number that represents the ratio of the new size to the original size. It is used to determine the size of the dilated figure.

Q: How do I dilate a line segment?

A: To dilate a line segment, you need to multiply the coordinates of the endpoints by the scale factor.

Q: What is the formula for dilating a line segment?

A: The formula for dilating a line segment is:

A' = (A_x * scale_factor, A_y * scale_factor) B' = (B_x * scale_factor, B_y * scale_factor)

where A' and B' are the dilated endpoints, A_x and A_y are the x and y coordinates of the original endpoint A, B_x and B_y are the x and y coordinates of the original endpoint B, and scale_factor is the scale factor.

Q: Can I dilate a line segment by a negative scale factor?

A: Yes, you can dilate a line segment by a negative scale factor. However, the direction of the line segment will be reversed.

Q: What happens if I dilate a line segment by a scale factor of 0?

A: If you dilate a line segment by a scale factor of 0, the line segment will be collapsed to a point.

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

A: Yes, you can dilate a line segment by a scale factor of 1. In this case, the line segment will remain unchanged.

Q: How do I find the midpoint of a dilated line segment?

A: To find the midpoint of a dilated line segment, you need to find the midpoint of the original line segment and then multiply it by the scale factor.

Q: Can I dilate a line segment by a scale factor that is not a whole number?

A: Yes, you can dilate a line segment by a scale factor that is not a whole number. However, the coordinates of the dilated endpoints will be decimal numbers.

Conclusion

In this article, we answered some frequently asked questions about dilating line segments. We hope that this article has been helpful in understanding the concept of dilation and how to dilate line segments.

Additional Resources

• Dilation of Line Segments
• Dilation of Points
• Dilation of Shapes

Practice Problems

1. Dilate the line segment with endpoints (2, 3) and (4, 5) by a scale factor of 2.
2. Dilate the line segment with endpoints (1, 2) and (3, 4) by a scale factor of 3.
3. Dilate the line segment with endpoints (0, 0) and (1, 1) by a scale factor of 4.
4. Dilate the line segment with endpoints (2, 2) and (3, 3) by a scale factor of 5.
5. Dilate the line segment with endpoints (1, 1) and (2, 2) by a scale factor of 6.

Answer Key

1. (4, 6) and (8, 10)
2. (3, 6) and (9, 12)
3. (0, 0) and (4, 4)
4. (10, 10) and (15, 15)
5. (7, 7) and (12, 12)