The coordinates of a triangle are described by a matrix, where the rows represent each point, A, B, and C, from top row to bottom row, and column 1 represents the x coordinates and column 2 represents the y coordinates. What transformation does the following matrix represent when added to the first matrix?
A. A rotation about the origin clockwise by 90°
B. A flip over the y-axis
C. A translation to the left by 20 units and down by 20 units
D. A translation to the right by 20 units and down by 20 units
The given matrix is:
$$
\begin{bmatrix}
-20 & -10 \\
-10 & 0 \\
\end{bmatrix}
$$
To understand the transformation represented by this matrix, we need to add it to the matrix representing the original triangle. Let's call the original matrix $M$:
$$
M = \begin{bmatrix}
x_A & y_A \\
x_B & y_B \\
x_C & y_C \\
\end{bmatrix}
$$
To add the given matrix to $M$, we add the corresponding entries:
$$
\begin{bmatrix}
x_A - 20 & y_A - 10 \\
x_B - 20 & y_B - 10 \\
x_C - 20 & y_C - 10 \\
\end{bmatrix}
$$
This means that all the points of the original triangle have been shifted to the left by 20 units and down by 10 units. Therefore, the transformation represented by the given matrix is a translation to the left by 20 units and down by 10 units.
So the answer is (C) a translation to the left by 20 units and down by 10 units.
To determine the transformation represented by the second matrix, we will compare the changes in the coordinates.
Let's assume the first matrix is:
| A B C |
|---|---|---|
| x y |
| x y |
| x y |
And the second matrix is:
| d e f |
|---|---|---|
| g h |
| g h |
| g h |
To find the transformation, we can compare the changes in each coordinate:
For point A:
- The x-coordinate changes from x to (x + d)
- The y-coordinate changes from y to (y + g)
For point B:
- The x-coordinate changes from x to (x + e)
- The y-coordinate changes from y to (y + h)
For point C:
- The x-coordinate changes from x to (x + f)
- The y-coordinate changes from y to (y + h)
Now, let's analyze the changes:
1. A rotation about the origin clockwise by 90°:
- In this transformation, the x-coordinate and y-coordinate change but not in this specific pattern.
2. A flip over the y-axis:
- In this transformation, the x-coordinate changes sign (positive to negative or vice versa), and the y-coordinate remains the same. However, this pattern is not observed.
3. A translation to the left by 20 units and down by 20 units:
- In this transformation, the x-coordinate decreases by 20 units, and the y-coordinate decreases by 20 units. This pattern is not observed.
4. A translation to the right by 20 units and down by 20 units:
- In this transformation, the x-coordinate increases by 20 units, and the y-coordinate decreases by 20 units. The second matrix follows this pattern, so we can conclude that the transformation represented by the second matrix is a translation to the right by 20 units and down by 20 units.
Therefore, the correct answer is D. A translation to the right by 20 units and down by 20 units.
To determine the transformation represented by the given matrix when added to the first matrix representing the coordinates of a triangle, we need to understand the effects of each operation on a point in the coordinate plane.
Let's represent the given transformation matrix as T:
| a b |
T = | c d |
To apply the transformation, we will multiply this matrix by each point's coordinates using matrix multiplication.
Let's assume the original matrix representing the triangle's coordinates is M:
| x1 y1 |
M = | x2 y2 |
| x3 y3 |
To calculate the result of adding the transformation matrix T to the original matrix M, the resulting matrix R will be:
R = M + T
To compute R, we will add the corresponding elements of M and T:
| x1 + a y1 + b |
R = | x2 + c y2 + d |
| x3 + c y3 + d |
Now, let's analyze each option to determine which transformation is represented:
A. A rotation about the origin clockwise by 90°
A rotation by 90° clockwise can be represented by the transformation matrix:
| 0 1 |
| -1 0 |
Comparing this with our given transformation matrix T = | a b | = | 0 1 |, we can see that a = 0 and b = 1. However, c and d are not matching the values for a rotation by 90°. Therefore, option A is not the correct transformation.
B. A flip over the y-axis
A flip over the y-axis can be represented by the transformation matrix:
| -1 0 |
| 0 1 |
Comparing this with our given transformation matrix T = | a b | = | -1 0 |, we see that a = -1 and b = 0. The other values of c and d do not match the values for a flip over the y-axis. Therefore, option B is not the correct transformation.
C. A translation to the left by 20 units and down by 20 units
A translation to the left by 20 units and down by 20 units will affect the x and y coordinates of each point.
Comparing this with our given transformation matrix T = | a b |, we want x1 + a = x1 - 20 and y1 + b = y1 - 20 to hold true for all points. Therefore, a = -20 and b = -20.
Checking the other values with similar calculations, we find that c = -20 and d = -20 as well. So, the transformation matrix matches the given values for translation to the left by 20 units and down by 20 units. Therefore, option C is the correct transformation.
D. A translation to the right by 20 units and down by 20 units
The given transformation matrix does not match the requirements for a translation to the right by 20 units and down by 20 units. Therefore, option D is not the correct transformation.
In conclusion, the correct transformation represented by the given matrix when added to the first matrix is option C: A translation to the left by 20 units and down by 20 units.