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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.

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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: \( L''''(7, 0), M'(8, 0), N(4, 2) \)