Match Each Quadrilateral, Described By Its Vertices, To The Sequence Of Transformations That Will Show It Is Congruent To Quadrilateral JKLM, Which Has Vertices J(8, 4), K(4, 10), L(12, 12), And M(14, 10).1. Quadrilateral S(4, 16), T(10, 20), U(12,

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Introduction

In geometry, congruent figures are those that have the same size and shape. To prove that two quadrilaterals are congruent, we need to show that they have the same size and shape. One way to do this is by applying a sequence of transformations to one of the quadrilaterals until it matches the other. In this article, we will explore how to match each quadrilateral, described by its vertices, to the sequence of transformations that will show it is congruent to quadrilateral JKLM.

Understanding Quadrilateral JKLM

Quadrilateral JKLM has vertices J(8, 4), K(4, 10), L(12, 12), and M(14, 10). To understand its shape and size, we can plot its vertices on a coordinate plane.

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| J | 8 | 4 |
| K | 4 | 10 |
| L | 12 | 12 |
| M | 14 | 10 |

Transformation 1: Translation

A translation is a transformation that moves a figure from one location to another without changing its size or shape. To apply a translation, we need to specify the direction and distance of the movement.

Example 1: Translation of Quadrilateral ABCD

Suppose we have a quadrilateral ABCD with vertices A(2, 2), B(4, 2), C(4, 4), and D(2, 4). To translate it 3 units to the right and 2 units up, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| A | 2 | 2 |
| B | 4 | 2 |
| C | 4 | 4 |
| D | 2 | 4 |





New Vertex
x-coordinate
y-coordinate






A
5
4




B
7
4




C
7
6




D
5
6

Example 2: Translation of Quadrilateral EFGH

Suppose we have a quadrilateral EFGH with vertices E(6, 6), F(8, 6), G(8, 8), and H(6, 8). To translate it 2 units to the left and 1 unit down, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| E | 6 | 6 |
| F | 8 | 6 |
| G | 8 | 8 |
| H | 6 | 8 |





New Vertex
x-coordinate
y-coordinate






E
4
5




F
6
5




G
6
7




H
4
7

Transformation 2: Rotation

A rotation is a transformation that turns a figure around a fixed point without changing its size or shape. To apply a rotation, we need to specify the angle of rotation and the center of rotation.

Example 1: Rotation of Quadrilateral IJKL

Suppose we have a quadrilateral IJKL with vertices I(10, 10), J(12, 10), K(12, 12), and L(10, 12). To rotate it 90 degrees clockwise around the origin, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| I | 10 | 10 |
| J | 12 | 10 |
| K | 12 | 12 |
| L | 10 | 12 |





New Vertex
x-coordinate
y-coordinate






I
10
-10




J
10
-12




K
-12
-10




L
-10
-12

Example 2: Rotation of Quadrilateral MNOP

Suppose we have a quadrilateral MNOP with vertices M(14, 14), N(16, 14), O(16, 16), and P(14, 16). To rotate it 180 degrees around the origin, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| M | 14 | 14 |
| N | 16 | 14 |
| O | 16 | 16 |
| P | 14 | 16 |





New Vertex
x-coordinate
y-coordinate






M
-14
-14




N
-16
-14




O
-16
-16




P
-14
-16

Transformation 3: Reflection

A reflection is a transformation that flips a figure over a line without changing its size or shape. To apply a reflection, we need to specify the line of reflection.

Example 1: Reflection of Quadrilateral QRST

Suppose we have a quadrilateral QRST with vertices Q(18, 18), R(20, 18), S(20, 20), and T(18, 20). To reflect it over the x-axis, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| Q | 18 | 18 |
| R | 20 | 18 |
| S | 20 | 20 |
| T | 18 | 20 |





New Vertex
x-coordinate
y-coordinate






Q
18
-18




R
20
-18




S
20
-20




T
18
-20

Example 2: Reflection of Quadrilateral UVWX

Suppose we have a quadrilateral UVWX with vertices U(22, 22), V(24, 22), W(24, 24), and X(22, 24). To reflect it over the y-axis, we can apply the following transformation:

| Vertex | x-coordinate | y-coordinate |
| --- | --- | --- |
| U | 22 | 22 |
| V | 24 | 22 |
| W | 24 | 24 |
| X | 22 | 24 |





New Vertex
x-coordinate
y-coordinate






U
-22
22




V
-24
22




W
-24
24




X
-22
24

Conclusion

Introduction

In our previous article, we explored how to match each quadrilateral, described by its vertices, to the sequence of transformations that will show it is congruent to quadrilateral JKLM. In this article, we will answer some frequently asked questions about transformations of quadrilaterals.

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

A: A translation is a transformation that moves a figure from one location to another without changing its size or shape. A rotation is a transformation that turns a figure around a fixed point without changing its size or shape.

Q: How do I determine the direction and distance of a translation?

A: To determine the direction and distance of a translation, you need to specify the coordinates of the original and final positions of the figure. For example, if you want to translate a figure 3 units to the right and 2 units up, you can specify the coordinates of the original position as (x, y) and the coordinates of the final position as (x + 3, y + 2).

Q: How do I determine the angle of rotation?

A: To determine the angle of rotation, you need to specify the number of degrees the figure is rotated. For example, if you want to rotate a figure 90 degrees clockwise, you can specify the angle of rotation as 90°.

Q: How do I determine the center of rotation?

A: To determine the center of rotation, you need to specify the coordinates of the point around which the figure is rotated. For example, if you want to rotate a figure around the origin (0, 0), you can specify the center of rotation as (0, 0).

Q: What is the difference between a reflection over the x-axis and a reflection over the y-axis?

A: A reflection over the x-axis is a transformation that flips a figure over the x-axis without changing its size or shape. A reflection over the y-axis is a transformation that flips a figure over the y-axis without changing its size or shape.

Q: How do I determine the line of reflection?

A: To determine the line of reflection, you need to specify the equation of the line over which the figure is reflected. For example, if you want to reflect a figure over the x-axis, you can specify the equation of the line as y = 0.

Q: Can I apply multiple transformations to a figure?

A: Yes, you can apply multiple transformations to a figure. For example, you can first translate a figure 3 units to the right and 2 units up, and then rotate it 90 degrees clockwise.

Q: How do I determine if two figures are congruent?

A: To determine if two figures are congruent, you need to show that they have the same size and shape. You can do this by applying a sequence of transformations to one of the figures until it matches the other.

Q: What are some real-world applications of transformations of quadrilaterals?

A: Transformations of quadrilaterals have many real-world applications, such as:

  • Computer graphics: Transformations are used to create 3D models and animations.
  • Engineering: Transformations are used to design and analyze mechanical systems.
  • Architecture: Transformations are used to design and analyze buildings and other structures.
  • Science: Transformations are used to model and analyze physical systems.

Conclusion

In this article, we have answered some frequently asked questions about transformations of quadrilaterals. We hope this article has helped you understand the concepts of translation, rotation, and reflection, and how to apply them to solve problems. If you have any more questions, feel free to ask!