A triangle has vertices (-2,-3), (3,5), and (8,-1)
Write a matrix expression to represent rotating the triangle 90° clockwise. Evaluate the matrix expression and list the coordinates of the vertices of the transformed figure.
Please help me! I was absent for the entirety of this lesson due to medical reasons, and I have a test tomorrow!
a 90° clockwise rotation takes
(x,y) -> (y,-x)
review your rotation matrix definition.
| +1+0 | -2 +3 +8 |
| +0 -1 | -3 +5 -1 |
| -2 +3 +8 |
| +3 -5 +1 |
(-2,3) (3, -5) (8,1)
That above is for part a) of the assignment if clarification is needed...
Have a blessed day!
No problem! I'll explain to you how to rotate the triangle and then we can work on solving it together.
To rotate a point (x, y) in the coordinate plane 90 degrees clockwise, you can use the following matrix expression:
\[ \begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix} \begin{bmatrix}
x \\
y \\
\end{bmatrix} \]
Now, let's solve it step by step:
1. Write the matrix expression to represent rotating the triangle 90° clockwise.
To apply the matrix expression to each vertex of the triangle, we can represent the vertices as column vectors:
\[ \begin{bmatrix}
x' \\
y' \\
\end{bmatrix} = \begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix} \begin{bmatrix}
x \\
y \\
\end{bmatrix} \]
2. Evaluate the matrix expression.
For each vertex of the triangle, substitute the coordinates into the matrix expression:
Vertex 1:
\[ \begin{bmatrix}
x'_1 \\
y'_1 \\
\end{bmatrix} = \begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix} \begin{bmatrix}
-2 \\
-3 \\
\end{bmatrix} \]
Vertex 2:
\[ \begin{bmatrix}
x'_2 \\
y'_2 \\
\end{bmatrix} = \begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix} \begin{bmatrix}
3 \\
5 \\
\end{bmatrix} \]
Vertex 3:
\[ \begin{bmatrix}
x'_3 \\
y'_3 \\
\end{bmatrix} = \begin{bmatrix}
0 & 1 \\
-1 & 0 \\
\end{bmatrix} \begin{bmatrix}
8 \\
-1 \\
\end{bmatrix} \]
3. Simplify the matrix multiplication.
To multiply matrices, we multiply the corresponding elements and sum them up.
Vertex 1:
\[ \begin{bmatrix}
x'_1 \\
y'_1 \\
\end{bmatrix} = \begin{bmatrix}
0 \\
-1 \\
\end{bmatrix} \]
Vertex 2:
\[ \begin{bmatrix}
x'_2 \\
y'_2 \\
\end{bmatrix} = \begin{bmatrix}
5 \\
-3 \\
\end{bmatrix} \]
Vertex 3:
\[ \begin{bmatrix}
x'_3 \\
y'_3 \\
\end{bmatrix} = \begin{bmatrix}
-1 \\
8 \\
\end{bmatrix} \]
4. Write the coordinates of the transformed figure.
The transformed figure has the following vertices:
Vertex 1: (-1, -1)
Vertex 2: (5, -3)
Vertex 3: (-3, 8)
So, the coordinates of the vertices of the transformed figure are (-1, -1), (5, -3), and (-3, 8).
I hope this explanation helps! Let me know if you have any further questions.