Question

vertices Upper L left parenthesis 3 comma negative 2 right parenthesisL(3,−2)​, Upper M left parenthesis 7 comma negative 3 right parenthesisM(7,−3)​, and Upper N left parenthesis 5 comma 4 right parenthesisN(5,4). Find the coordinates of the vertices after a 90degrees° rotation about the origin and about each of the points​ L, M, and N.

Answer 1

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:

1. **Rotation about the Origin**:
- A point \((x, y)\) after a 90° counterclockwise rotation about the origin transforms to \((-y, x)\).

2. **Rotation about a Point \((a, b)\)**:
- To rotate the point \((x, y)\) about another point \((a, b)\), we:
1. Translate the point so that \((a, b)\) becomes the origin.
2. Rotate around the origin.
3. Translate back.

### 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 \(M(7, -3)\):
1. Translate: \(M' = (7 - 3, -3 + 2) = (4, -1)\)
2. Rotate about origin: \((-(-1), 4) = (1, 4)\)
3. Translate back: \((1 + 3, 4 - 2) = (4, 2)\)

#### For \(N(5, 4)\):
1. Translate: \(N' = (5 - 3, 4 + 2) = (2, 6)\)
2. Rotate about origin: \((-6, 2) = (-6, 2)\)
3. 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 \(L(3, -2)\):
1. Translate: \(L' = (3 - 7, -2 + 3) = (-4, 1)\)
2. Rotate about origin: \((-1, -4) = (-1, -4)\)
3. Translate back: \((-1 + 7, -4 - 3) = (6, -7)\)

#### For \(N(5, 4)\):
1. Translate: \(N' = (5 - 7, 4 + 3) = (-2, 7)\)
2. Rotate about origin: \((-7, -2) = (-7, -2)\)
3. 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 \(L(3, -2)\):
1. Translate: \(L' = (3 - 5, -2 - 4) = (-2, -6)\)
2. Rotate about origin: \((6, -2) = (6, -2)\)
3. Translate back: \((6 + 5, -2 + 4) = (11, 2)\)

#### For \(M(7, -3)\):
1. Translate: \(M' = (7 - 5, -3 - 4) = (2, -7)\)
2. Rotate about origin: \((7, 2) = (7, 2)\)
3. 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**:
- \(L'' = (11, 2)\)
- \(M'' = (12, 6)\)