Enlargement with Scale Factor 3

Oct 13, 2025 by windailys.com 83 views

Introduction

In the realm of mathematics, geometry plays a vital role in understanding the properties and transformations of shapes. One such shape is the image of LOVE, represented by the vertices L(2,2), O(6,2), V(7,4), and E(3,4). In this article, we will embark on a mathematical journey to find the coordinates of the vertices of the image after a positive 90° rotation about the origin, followed by an enlargement with center (0,0) and scale factor of 3.

Rotation and Enlargement: A Mathematical Framework

To tackle this problem, we need to understand the concepts of rotation and enlargement in mathematics. A rotation is a transformation that turns a shape around a fixed point, known as the center of rotation. In this case, the center of rotation is the origin (0,0). An enlargement, on the other hand, is a transformation that changes the size of a shape while maintaining its shape and orientation.

90° Rotation about the Origin

A 90° rotation about the origin can be represented by the following matrix:

\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}

This matrix rotates a point (x,y) to (-y,x). Let's apply this rotation to the vertices of the image of LOVE:

L(2,2) → L'(-2,2)
O(6,2) → O'(-2,6)
V(7,4) → V'(-4,7)
E(3,4) → E'(-4,3)

Enlargement with Scale Factor 3

An enlargement with scale factor 3 can be represented by the following matrix:

\begin{bmatrix}
3 & 0 \\
0 & 3
\end{bmatrix}

This matrix enlarges a point (x,y) to (3x,3y). Let's apply this enlargement to the rotated vertices:

L'(-2,2) → L''(-6,6)
O'(-2,6) → O''(-6,18)
V'(-4,7) → V''(-12,21)
E'(-4,3) → E''(-12,9)

Conclusion

In this article, we have explored the mathematical concepts of rotation and enlargement to find the coordinates of the vertices of the image of LOVE after a positive 90° rotation about the origin, followed by an enlargement with center (0,0) and scale factor of 3. By applying the rotation and enlargement matrices, we have obtained the new coordinates of the vertices:

L''(-6,6)
O''(-6,18)
V''(-12,21)
E''(-12,9)

Introduction

In our previous article, we explored the mathematical concepts of rotation and enlargement to find the coordinates of the vertices of the image of LOVE after a positive 90° rotation about the origin, followed by an enlargement with center (0,0) and scale factor of 3. In this article, we will address some of the frequently asked questions related to this topic.

Q&A

Q: What is the center of rotation in this problem?

A: The center of rotation is the origin (0,0).

Q: What is the scale factor for the enlargement?

A: The scale factor for the enlargement is 3.

Q: How do I apply the rotation matrix to the vertices of the image of LOVE?

A: To apply the rotation matrix, you need to multiply the rotation matrix by the coordinates of each vertex. For example, to rotate the vertex L(2,2), you would multiply the rotation matrix by the coordinates (2,2) to get the new coordinates L'(-2,2).

Q: How do I apply the enlargement matrix to the rotated vertices?

A: To apply the enlargement matrix, you need to multiply the enlargement matrix by the coordinates of each rotated vertex. For example, to enlarge the rotated vertex L'(-2,2), you would multiply the enlargement matrix by the coordinates (-2,2) to get the new coordinates L''(-6,6).

Q: What is the significance of the scale factor in this problem?

A: The scale factor determines the amount of enlargement applied to the rotated vertices. A scale factor of 3 means that each coordinate of the rotated vertex is multiplied by 3 to get the new coordinates.

Q: Can I apply the rotation and enlargement transformations in any order?

A: No, the rotation and enlargement transformations must be applied in the order specified: first, a 90° rotation about the origin, and then an enlargement with center (0,0) and scale factor of 3.

Q: How do I find the coordinates of the vertices of the image of LOVE after multiple transformations?

A: To find the coordinates of the vertices after multiple transformations, you need to apply each transformation in sequence, using the new coordinates as the input for the next transformation.

Q: Can I use this method to find the coordinates of the vertices of any shape after rotation and enlargement?

A: Yes, this method can be used to find the coordinates of the vertices of any shape after rotation and enlargement, as long as the center of rotation and scale factor are specified.

Conclusion

In this article, we have addressed some of the frequently asked questions related to the mathematical concepts of rotation and enlargement. We hope that this Q&A article has provided a better understanding of the topic and has helped to clarify any doubts. If you have any further questions, please feel free to ask.