Question

Draw Upper DeltaΔLMN with vertices Upper L left parenthesis 2 comma negative 1 right parenthesisL(2,−1)​, Upper M left parenthesis 6 comma negative 2 right parenthesisM(6,−2)​, and Upper N left parenthesis 4 comma 2 right parenthesisN(4,2). 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 of triangle \( \Delta LMN \) after a 90° rotation about the origin and about each of the points \( L \), \( M \), and \( N \), we can follow these steps:

1. **90° Rotation about the Origin**:
The formula for rotating a point \( (x, y) \) by 90° counterclockwise around the origin is \( (-y, x) \).

- For \( L(2, -1) \):
\[
L' = (-(-1), 2) = (1, 2)
\]
- For \( M(6, -2) \):
\[
M' = (-(-2), 6) = (2, 6)
\]
- For \( N(4, 2) \):
\[
N' = (-2, 4) = (-2, 4)
\]

So the new coordinates after a 90° rotation about the origin are:
\[
L'(1, 2), \quad M'(2, 6), \quad N'(-2, 4)
\]

2. **90° Rotation about Point L(2, -1)**:
To rotate about point \( L(2, -1) \), we first need to translate the triangle such that \( L \) is at the origin, perform the rotation, and then translate back.

New points relative to \( L(2, -1) \):
- \( L \) relative to \( L \) is \( (0, 0) \)
- \( M(6, -2) \) relative to \( L \):
\[
(6 - 2, -2 + 1) = (4, -1)
\]
- \( N(4, 2) \) relative to \( L \):
\[
(4 - 2, 2 + 1) = (2, 3)
\]

Now we rotate these points 90° around the origin:
- For \( M(4, -1) \):
\[
M'' = (-(-1), 4) = (1, 4)
\]
- For \( N(2, 3) \):
\[
N'' = (-3, 2) = (-3, 2)
\]

Now we translate back by adding \( L(2, -1) \):
- \( M''(1, 4) + (2, -1) = (3, 3) \)
- \( N''(-3, 2) + (2, -1) = (-1, 1) \)

So the new coordinates after a 90° rotation about point \( L \) are:
\[
L(2, -1), \quad M''(3, 3), \quad N''(-1, 1)
\]

3. **90° Rotation about Point M(6, -2)**:
Similar steps:
- \( L(2, -1) \) relative to \( M \):
\[
(2 - 6, -1 + 2) = (-4, 1)
\]
- \( N(4, 2) \) relative to \( M \):
\[
(4 - 6, 2 + 2) = (-2, 4)
\]

After a 90° rotation:
- For \( L(-4, 1) \):
\[
L''' = (-1, -4) = (-1, -4)
\]
- For \( N(-2, 4) \):
\[
N''' = (-4, -2) = (-4, -2)
\]

Translate back by adding \( M(6, -2) \):
- \( L'''(-1, -4) + (6, -2) = (5, -6) \)
- \( N'''(-4, -2) + (6, -2) = (2, -4) \)

So the new coordinates after a 90° rotation about point \( M \) are:
\[
L'''(5, -6), \quad M(6, -2), \quad N'''(2, -4)
\]

4. **90° Rotation about Point N(4, 2)**:
Process this similarly:
- \( L(2, -1) \) relative to \( N \):
\[
(2 - 4, -1 - 2) = (-2, -3)
\]
- \( M(6, -2) \) relative to \( N \):
\[
(6 - 4, -2 - 2) = (2, -4)
\]

After a 90° rotation:
- For \( L(-2, -3) \):
\[
L'''' = (3, -2) = (3, -2)
\]
- For \( M(2, -4) \):
\[
M'''' = (4, -2) = (4, -2)
\]

Translate back by adding \( N(4, 2) \):
- \( L''''(3, -2) + (4, 2) = (7, 0) \)
- \( M''''(4, -2) + (4, 2) = (8, 0) \)

So the new coordinates after a 90° rotation about point \( N \) are:
\[
L''''(7, 0), \quad M'(8, 0), \quad N(4, 2)
\]

In summary, the coordinates after the rotations are:
- 90° rotation about the **origin**: \( L'(1, 2), M'(2, 6), N'(-2, 4) \)
- 90° rotation about **L**: \( L(2, -1), M''(3, 3), N''(-1, 1) \)
- 90° rotation about **M**: \( L'''(5, -6), M(6, -2), N'''(2, -4) \)
- 90° rotation about **N**: \( L''''(7, 0), M'(8, 0), N(4, 2) \)