To find the coordinates of the vertices after a 90° rotation about the origin and about each of the points L, M, and N, we will use the following rules for rotation:
-
Rotation about the Origin:
- A point \((x, y)\) after a 90° counterclockwise rotation about the origin transforms to \((-y, x)\).
-
Rotation about a Point \((a, b)\):
- To rotate the point \((x, y)\) about another point \((a, b)\), we:
- Translate the point so that \((a, b)\) becomes the origin.
- Rotate around the origin.
- Translate back.
- To rotate the point \((x, y)\) about another point \((a, b)\), we:
Given Points:
- \(L(3, -2)\)
- \(M(7, -3)\)
- \(N(5, 4)\)
1. Rotation about the Origin:
For \(L(3, -2)\):
- New coordinates: \((-(-2), 3) = (2, 3)\)
For \(M(7, -3)\):
- New coordinates: \(-(-3), 7) = (3, 7)\)
For \(N(5, 4)\):
- New coordinates: \((-4, 5) = (-4, 5)\)
Coordinates after 90° rotation about the Origin:
- \(L' = (2, 3)\)
- \(M' = (3, 7)\)
- \(N' = (-4, 5)\)
2. Rotation about Point \(L(3, -2)\):
For \(M(7, -3)\):
- Translate: \(M' = (7 - 3, -3 + 2) = (4, -1)\)
- Rotate about origin: \((-(-1), 4) = (1, 4)\)
- Translate back: \((1 + 3, 4 - 2) = (4, 2)\)
For \(N(5, 4)\):
- Translate: \(N' = (5 - 3, 4 + 2) = (2, 6)\)
- Rotate about origin: \((-6, 2) = (-6, 2)\)
- Translate back: \((-6 + 3, 2 - 2) = (-3, 0)\)
Coordinates after 90° rotation about \(L\):
- \(M'' = (4, 2)\)
- \(N'' = (-3, 0)\)
3. Rotation about Point \(M(7, -3)\):
For \(L(3, -2)\):
- Translate: \(L' = (3 - 7, -2 + 3) = (-4, 1)\)
- Rotate about origin: \((-1, -4) = (-1, -4)\)
- Translate back: \((-1 + 7, -4 - 3) = (6, -7)\)
For \(N(5, 4)\):
- Translate: \(N' = (5 - 7, 4 + 3) = (-2, 7)\)
- Rotate about origin: \((-7, -2) = (-7, -2)\)
- Translate back: \((-7 + 7, -2 - 3) = (0, -5)\)
Coordinates after 90° rotation about \(M\):
- \(L'' = (6, -7)\)
- \(N'' = (0, -5)\)
4. Rotation about Point \(N(5, 4)\):
For \(L(3, -2)\):
- Translate: \(L' = (3 - 5, -2 - 4) = (-2, -6)\)
- Rotate about origin: \((6, -2) = (6, -2)\)
- Translate back: \((6 + 5, -2 + 4) = (11, 2)\)
For \(M(7, -3)\):
- Translate: \(M' = (7 - 5, -3 - 4) = (2, -7)\)
- Rotate about origin: \((7, 2) = (7, 2)\)
- Translate back: \((7 + 5, 2 + 4) = (12, 6)\)
Coordinates after 90° rotation about \(N\):
- \(L'' = (11, 2)\)
- \(M'' = (12, 6)\)
Summary of Rotated Coordinates:
About Origin:
- \(L' = (2, 3)\)
- \(M' = (3, 7)\)
- \(N' = (-4, 5)\)
About L:
- \(M'' = (4, 2)\)
- \(N'' = (-3, 0)\)
About M:
- \(L'' = (6, -7)\)
- \(N'' = (0, -5)\)
About N:
- \(L'' = (11, 2)\)
- \(M'' = (12, 6)\)