To solve for the coordinates of the vertices of triangle [tex]\( MNP \)[/tex] after it has been rotated by [tex]\(-270^\circ\)[/tex] about the origin, we need to follow specific steps for rotational transformation. A [tex]\(-270^\circ\)[/tex] rotation is equivalent to a [tex]\(90^\circ\)[/tex] clockwise rotation.
The rule for rotating a point [tex]\((x, y)\)[/tex] by [tex]\(90^\circ\)[/tex] clockwise is:
[tex]\[
(x, y) \rightarrow (y, -x)
\][/tex]
Now, let's apply this rule to each vertex of the triangle.
1. For vertex [tex]\( M(7, -2) \)[/tex]:
- The new coordinates will be obtained by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and changing the sign of the new x-coordinate:
[tex]\[
M(7, -2) \rightarrow M'(-2, -7)
\][/tex]
2. For vertex [tex]\( N(4, -3) \)[/tex]:
[tex]\[
N(4, -3) \rightarrow N'(-3, -4)
\][/tex]
3. For vertex [tex]\( P(2, -1) \)[/tex]:
[tex]\[
P(2, -1) \rightarrow P'(-1, -2)
\][/tex]
Thus, the coordinates of the vertices of the triangle after the [tex]\(-270^\circ\)[/tex] rotation are:
- Vertex [tex]\(M'\)[/tex] at [tex]\((-2, -7)\)[/tex]
- Vertex [tex]\(N'\)[/tex] at [tex]\((-3, -4)\)[/tex]
- Vertex [tex]\(P'\)[/tex] at [tex]\((-1, -2)\)[/tex]
So the final coordinates after rotation are:
[tex]\[
M (-2, -7) \\
N (-3, -4) \\
P (-1, -2)
\][/tex]
