The vertices of triangle LMN are L(3,1) M(4,3) N(-2,1). a transformation maps triangle onto one with vertices at L(-1,3) M(0,5) N(2,-1). Describe transformation and find its matrix
To describe the transformation that maps triangle LMN onto the one with vertices at L(-1,3), M(0,5), and N(2,-1), we can analyze the changes in the coordinates.
1. Translation: The triangle has been moved left by 4 units and up by 2 units.
- The x-coordinate of each vertex has been reduced by 4.
- The y-coordinate of each vertex has been increased by 2.
2. Reflection: The triangle has been reflected about the line y = x.
- The x-coordinate of each vertex becomes the y-coordinate in the new triangle.
- The y-coordinate of each vertex becomes the x-coordinate in the new triangle.
Now, let's find the transformation matrix for this transformation:
Let T represent the transformation matrix.
- For the translation part:
- The translation matrix is:
| 1 0 -4 |
| 0 1 2 |
| 0 0 1 |
- For the reflection part:
- The reflection matrix is:
| 0 1 0 |
| 1 0 0 |
| 0 0 1 |
To find the overall transformation matrix, we perform matrix multiplication between the translation and reflection matrices:
T = Translation Matrix * Reflection Matrix
T =
| 1 0 -4 | | 0 1 0 |
| 0 1 2 | * | 1 0 0 |
| 0 0 1 | | 0 0 1 |
Simplifying the multiplication gives us the final transformation matrix:
T =
| 0 1 0 |
| 1 0 2 |
| 0 0 1 |
Therefore, the transformation matrix for the given transformation is:
| 0 1 0 |
| 1 0 2 |
| 0 0 1 |
To describe the transformation that maps triangle LMN onto the one with vertices at L(-1,3), M(0,5), and N(2,-1), we can determine the translation and scaling involved.
Translation refers to the movement of an object without changing its shape or size. In this case, we can observe that the new triangle has been shifted 4 units to the left and 2 units up compared to the original triangle.
Scaling is the process of changing the size of an object. To find the scaling factor, we can compare the distances between the corresponding vertices of the two triangles.
The distance between L(3,1) and M(4,3) in the original triangle is calculated as follows:
√((4 - 3)^2 + (3 - 1)^2) = √(1 + 4) = √5
The distance between L(-1,3) and M(0,5) in the transformed triangle is:
√((0 - (-1))^2 + (5 - 3)^2) = √((1)^2 + 4) = √5
Similarly, we can compare the distances LM, MN, and NL to see if they are equal to √5. If they are, it means there is no scaling involved, and the transformation is only a translation.
Now, let's find the matrix representation of the transformation. For a translation, the matrix would be:
[1 0 tx]
[0 1 ty]
[0 0 1 ]
where tx is the translation in the x-axis (left or right) and ty is the translation in the y-axis (up or down).
From our observation, the triangle has been translated 4 units to the left and 2 units up. Therefore, the matrix representation of the translation is:
[1 0 -4]
[0 1 2]
[0 0 1 ]
To summarize, the transformation is a translation of -4 units along the x-axis (to the left) and a translation of 2 units along the y-axis (up). The matrix representation of this transformation is:
[1 0 -4]
[0 1 2]
[0 0 1 ]