To solve the problem of dilating a triangle with a given magnitude, let's follow the steps methodically.
### Step-by-Step Solution
1. Identify the coordinates of the original triangle vertices:
[tex]\[
\text{{Original triangle coordinates: }}
\begin{bmatrix}
3 & 6 & 3 \\
-3 & 3 & 3
\end{bmatrix}
\][/tex]
This matrix represents the coordinates [tex]\((3, -3)\)[/tex], [tex]\((6, 3)\)[/tex], and [tex]\((3, 3)\)[/tex].
2. Determine the magnitude of dilation:
The given magnitude for dilation is [tex]\( 3 \)[/tex].
3. Apply the dilation to each coordinate:
Dilation involves multiplying each coordinate by the given magnitude.
For the first vertex [tex]\((3, -3)\)[/tex]:
[tex]\[
3 \times 3 = 9 \quad \text{and} \quad -3 \times 3 = -9
\][/tex]
So the new coordinate is [tex]\((9, -9)\)[/tex].
For the second vertex [tex]\((6, 3)\)[/tex]:
[tex]\[
6 \times 3 = 18 \quad \text{and} \quad 3 \times 3 = 9
\][/tex]
So the new coordinate is [tex]\((18, 9)\)[/tex].
For the third vertex [tex]\((3, 3)\)[/tex]:
[tex]\[
3 \times 3 = 9 \quad \text{and} \quad 3 \times 3 = 9
\][/tex]
So the new coordinate is [tex]\((9, 9)\)[/tex].
4. Compile the new coordinates into a matrix:
After dilation, the new coordinates of the triangle vertices form the matrix:
[tex]\[
\begin{bmatrix}
9 & 18 & 9 \\
-9 & 9 & 9
\end{bmatrix}
\][/tex]
Thus, the dilated triangle coordinates are:
[tex]\[
\begin{bmatrix}
9 & 18 & 9 \\
-9 & 9 & 9
\end{bmatrix}
\][/tex]
### Conclusion:
The dilation of the given triangle by a magnitude of 3 results in the new triangle coordinates:
[tex]\[
\begin{bmatrix}
9 & 18 & 9 \\
-9 & 9 & 9
\end{bmatrix}
\][/tex]
