What Is The Scale Factor Of A Triangle With A Vertex At { A(-6, 4) $}$ That Has Been Dilated With A Center Of Dilation At The Origin, If The Vertex Of Its Image Is { A'(-24, 16) $}$?

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Introduction

In geometry, dilation is a transformation that changes the size of a figure. When a figure is dilated, its size increases or decreases, but its shape remains the same. The center of dilation is the point from which the dilation occurs, and the scale factor is the ratio of the distance between the center of dilation and the image of a point to the distance between the center of dilation and the original point. In this article, we will discuss how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16).

Understanding 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. The center of dilation is the point from which the dilation occurs, and the scale factor is the ratio of the distance between the center of dilation and the image of a point to the distance between the center of dilation and the original point. In other words, the scale factor is the factor by which the distance between the center of dilation and the original point is multiplied to get the distance between the center of dilation and the image of the point.

Finding the Scale Factor

To find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16), we need to use the formula for the scale factor. The formula for the scale factor is:

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point)

In this case, the center of dilation is the origin (0, 0), the original point is A(-6, 4), and the image of the point is A'(-24, 16). We can use the distance formula to find the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point.

Calculating the Distance

To calculate the distance between two points, we can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we need to find the distance between the center of dilation (0, 0) and the original point A(-6, 4), and the distance between the center of dilation (0, 0) and the image of the point A'(-24, 16).

Distance between center of dilation and original point = √((-6 - 0)^2 + (4 - 0)^2) Distance between center of dilation and original point = √((-6)^2 + 4^2) Distance between center of dilation and original point = √(36 + 16) Distance between center of dilation and original point = √52

Distance between center of dilation and image of point = √((-24 - 0)^2 + (16 - 0)^2) Distance between center of dilation and image of point = √((-24)^2 + 16^2) Distance between center of dilation and image of point = √(576 + 256) Distance between center of dilation and image of point = √832

Finding the Scale Factor

Now that we have the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point, we can use the formula for the scale factor to find the scale factor of the triangle.

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point) Scale factor = (√832) / (√52) Scale factor = (√(16*52)) / (√52) Scale factor = (√16 * √52) / (√52) Scale factor = √16 Scale factor = 4

Conclusion

In this article, we discussed how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16). We used the formula for the scale factor and calculated the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point. We then used the formula for the scale factor to find the scale factor of the triangle, which was 4.

Understanding the Importance of Scale Factor

The scale factor is an important concept in geometry, as it helps us understand how a figure changes in size when it is dilated. The scale factor is used to describe the ratio of the distance between the center of dilation and the image of a point to the distance between the center of dilation and the original point. In this article, we saw how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16).

Real-World Applications of Scale Factor

The scale factor has many real-world applications, including:

  • Architecture: Architects use the scale factor to design buildings and other structures that are proportional to the original design.
  • Engineering: Engineers use the scale factor to design machines and other devices that are proportional to the original design.
  • Art: Artists use the scale factor to create proportional drawings and paintings.
  • Computer Graphics: Computer graphics artists use the scale factor to create proportional images and animations.

Final Thoughts

In conclusion, the scale factor is an important concept in geometry that helps us understand how a figure changes in size when it is dilated. We discussed how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16). We used the formula for the scale factor and calculated the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point. We then used the formula for the scale factor to find the scale factor of the triangle, which was 4.

Introduction

In our previous article, we discussed how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16). In this article, we will answer some frequently asked questions about the scale factor of a triangle.

Q1: What is the scale factor of a triangle?

A1: The scale factor of a triangle is the ratio of the distance between the center of dilation and the image of a point to the distance between the center of dilation and the original point.

Q2: How do I find the scale factor of a triangle?

A2: To find the scale factor of a triangle, you need to use the formula for the scale factor, which is:

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point)

Q3: What is the center of dilation?

A3: The center of dilation is the point from which the dilation occurs. In the case of a triangle, the center of dilation is usually the origin (0, 0).

Q4: How do I calculate the distance between two points?

A4: To calculate the distance between two points, you can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Q5: What is the scale factor of a triangle with a vertex at A(2, 3) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-6, -9)?

A5: To find the scale factor of this triangle, we need to use the formula for the scale factor. First, we need to calculate the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point.

Distance between center of dilation and original point = √((2 - 0)^2 + (3 - 0)^2) Distance between center of dilation and original point = √(2^2 + 3^2) Distance between center of dilation and original point = √(4 + 9) Distance between center of dilation and original point = √13

Distance between center of dilation and image of point = √((-6 - 0)^2 + (-9 - 0)^2) Distance between center of dilation and image of point = √((-6)^2 + (-9)^2) Distance between center of dilation and image of point = √(36 + 81) Distance between center of dilation and image of point = √117

Now, we can use the formula for the scale factor to find the scale factor of the triangle.

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point) Scale factor = (√117) / (√13) Scale factor = (√(9*13)) / (√13) Scale factor = (√9 * √13) / (√13) Scale factor = √9 Scale factor = 3

Q6: What is the scale factor of a triangle with a vertex at A(-4, 2) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-16, 8)?

A6: To find the scale factor of this triangle, we need to use the formula for the scale factor. First, we need to calculate the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point.

Distance between center of dilation and original point = √((-4 - 0)^2 + (2 - 0)^2) Distance between center of dilation and original point = √((-4)^2 + 2^2) Distance between center of dilation and original point = √(16 + 4) Distance between center of dilation and original point = √20

Distance between center of dilation and image of point = √((-16 - 0)^2 + (8 - 0)^2) Distance between center of dilation and image of point = √((-16)^2 + 8^2) Distance between center of dilation and image of point = √(256 + 64) Distance between center of dilation and image of point = √320

Now, we can use the formula for the scale factor to find the scale factor of the triangle.

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point) Scale factor = (√320) / (√20) Scale factor = (√(16*20)) / (√20) Scale factor = (√16 * √20) / (√20) Scale factor = √16 Scale factor = 4

Q7: What is the scale factor of a triangle with a vertex at A(5, 1) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-20, -4)?

A7: To find the scale factor of this triangle, we need to use the formula for the scale factor. First, we need to calculate the distance between the center of dilation and the original point and the distance between the center of dilation and the image of the point.

Distance between center of dilation and original point = √((5 - 0)^2 + (1 - 0)^2) Distance between center of dilation and original point = √(5^2 + 1^2) Distance between center of dilation and original point = √(25 + 1) Distance between center of dilation and original point = √26

Distance between center of dilation and image of point = √((-20 - 0)^2 + (-4 - 0)^2) Distance between center of dilation and image of point = √((-20)^2 + (-4)^2) Distance between center of dilation and image of point = √(400 + 16) Distance between center of dilation and image of point = √416

Now, we can use the formula for the scale factor to find the scale factor of the triangle.

Scale factor = (distance between center of dilation and image of point) / (distance between center of dilation and original point) Scale factor = (√416) / (√26) Scale factor = (√(16*26)) / (√26) Scale factor = (√16 * √26) / (√26) Scale factor = √16 Scale factor = 4

Conclusion

In this article, we answered some frequently asked questions about the scale factor of a triangle. We discussed how to find the scale factor of a triangle with a vertex at A(-6, 4) that has been dilated with a center of dilation at the origin, if the vertex of its image is A'(-24, 16). We also answered questions about the scale factor of triangles with different vertices and images. We hope that this article has been helpful in understanding the concept of the scale factor of a triangle.