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    Mathematics
    Grade9
    Easy

    Question

    Use matrix multiplication to find the image. Graph the polygon and its image.
    Reflect open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 3 2 end table close square brackets in the x-axis.

    hintHint:

    Find the Image matrix and then plot them on a graph..

    The correct answer is:

      * In Geometry, a reflection is known as a flip. A reflection is a mirror image of the shape. An image will reflect through a line, known as the line of reflection.
      *It is a type of transformation that produces mirror image of the shape.
      Given That:
      Use matrix multiplication to find the image. Graph the polygon and its image.
      Reflect open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 3 2 end table close square brackets in the x-axis.
      >>>Let (a, b) be a point in the plane. Then, the points of reflection with respect to x-axis is (a,-b).
      * We know that reflection matrix in the x-axis: open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets
      * Multiply the polygon matrix by the matrix for a reflection on the x-axis
      open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 3 2 end table close square brackets equals open square brackets table row cell 1 left parenthesis negative 4 right parenthesis plus 0 left parenthesis 0 right parenthesis end cell cell 1 left parenthesis negative 3 right parenthesis plus 0 left parenthesis 3 right parenthesis end cell cell 1 left parenthesis negative 1 right parenthesis plus 0 left parenthesis 2 right parenthesis end cell row cell 0 left parenthesis negative 4 right parenthesis plus left parenthesis negative 1 right parenthesis 0 end cell cell 0 left parenthesis negative 3 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis 3 right parenthesis end cell cell 0 left parenthesis negative 1 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis end cell end table close square brackets equals open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 cell negative 3 end cell cell negative 2 end cell end table close square brackets
      >>>The required graph is:

      Therefore, the required matrix is:
      open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 3 2 end table close square brackets equals open square brackets table row cell 1 left parenthesis negative 4 right parenthesis plus 0 left parenthesis 0 right parenthesis end cell cell 1 left parenthesis negative 3 right parenthesis plus 0 left parenthesis 3 right parenthesis end cell cell 1 left parenthesis negative 1 right parenthesis plus 0 left parenthesis 2 right parenthesis end cell row cell 0 left parenthesis negative 4 right parenthesis plus left parenthesis negative 1 right parenthesis 0 end cell cell 0 left parenthesis negative 3 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis 3 right parenthesis end cell cell 0 left parenthesis negative 1 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis end cell end table close square brackets equals open square brackets table row cell negative 4 end cell cell negative 3 end cell cell negative 1 end cell row 0 cell negative 3 end cell cell negative 2 end cell end table close square brackets

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      The matrix open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets show the reflection in

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      space space space space A space space space space B space space    C space space space space space D open square brackets table row cell negative 3 end cell 1 cell negative 2 end cell cell negative 4 end cell row 2 cell negative 1 end cell cell negative 2 end cell cell negative 1 end cell end table close square brackets

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      space space space space A space space space space B space space    C space space space space space D open square brackets table row cell negative 3 end cell 1 cell negative 2 end cell cell negative 4 end cell row 2 cell negative 1 end cell cell negative 2 end cell cell negative 1 end cell end table close square brackets

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      space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space A    B    C M a t r i x space o f space t h e space p o l y g o n space equals open square brackets table row cell negative 6 end cell cell negative 5 end cell cell negative 3 end cell row cell negative 4 end cell cell negative 1 end cell cell negative 2 end cell end table close square brackets

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      space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space A    B    C M a t r i x space o f space t h e space p o l y g o n space equals open square brackets table row cell negative 6 end cell cell negative 5 end cell cell negative 3 end cell row cell negative 4 end cell cell negative 1 end cell cell negative 2 end cell end table close square brackets

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      Use matrix multiplication to find the image. Graph the polygon and its image.
      Reflect open square brackets table attributes columnalign center center right center end attributes row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets in the y-axis.

      The required matrix for the image is:
      open square brackets table row cell negative 1 end cell 0 row 0 1 end table close square brackets open square brackets table row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets equals open square brackets table row cell left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis plus 0 left parenthesis negative 2 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 6 right parenthesis plus 0 left parenthesis negative 3 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 5 right parenthesis plus 0 left parenthesis negative 8 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis plus 0 left parenthesis negative 5 right parenthesis end cell row cell 0 left parenthesis 2 right parenthesis plus 1 left parenthesis negative 2 right parenthesis end cell cell 0 left parenthesis 6 right parenthesis plus 1 left parenthesis negative 3 right parenthesis end cell cell 0 left parenthesis 5 right parenthesis plus 1 left parenthesis negative 8 right parenthesis end cell cell 0 left parenthesis 2 right parenthesis plus 1 left parenthesis negative 5 right parenthesis end cell end table close square brackets equals open square brackets table attributes columnalign center center right left end attributes row cell negative 2 end cell cell negative 6 end cell cell negative 5 end cell cell negative 2 end cell row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets

      Use matrix multiplication to find the image. Graph the polygon and its image.
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      open square brackets table row cell negative 1 end cell 0 row 0 1 end table close square brackets open square brackets table row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets equals open square brackets table row cell left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis plus 0 left parenthesis negative 2 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 6 right parenthesis plus 0 left parenthesis negative 3 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 5 right parenthesis plus 0 left parenthesis negative 8 right parenthesis end cell cell left parenthesis negative 1 right parenthesis left parenthesis 2 right parenthesis plus 0 left parenthesis negative 5 right parenthesis end cell row cell 0 left parenthesis 2 right parenthesis plus 1 left parenthesis negative 2 right parenthesis end cell cell 0 left parenthesis 6 right parenthesis plus 1 left parenthesis negative 3 right parenthesis end cell cell 0 left parenthesis 5 right parenthesis plus 1 left parenthesis negative 8 right parenthesis end cell cell 0 left parenthesis 2 right parenthesis plus 1 left parenthesis negative 5 right parenthesis end cell end table close square brackets equals open square brackets table attributes columnalign center center right left end attributes row cell negative 2 end cell cell negative 6 end cell cell negative 5 end cell cell negative 2 end cell row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets

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      Use matrix multiplication to find the image. Graph the polygon and its image.
      Reflect open square brackets table attributes columnalign center center right center end attributes row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets in the x-axis.

      The matrix representation of the image :
      open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets open square brackets table row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets equals open square brackets table row cell 1 left parenthesis 2 right parenthesis plus 0 left parenthesis negative 2 right parenthesis end cell cell 1 left parenthesis 6 right parenthesis plus 0 left parenthesis negative 3 right parenthesis end cell cell 1 left parenthesis 5 right parenthesis plus 0 left parenthesis negative 8 right parenthesis end cell cell 1 left parenthesis 2 right parenthesis plus 0 left parenthesis negative 5 right parenthesis end cell row cell 0 left parenthesis 2 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 2 right parenthesis end cell cell 0 left parenthesis 6 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 3 right parenthesis end cell cell 0 left parenthesis 5 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 8 right parenthesis end cell cell 0 left parenthesis 2 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 5 right parenthesis end cell end table close square brackets equals open square brackets table attributes columnalign center center right left end attributes row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets

      Use matrix multiplication to find the image. Graph the polygon and its image.
      Reflect open square brackets table attributes columnalign center center right center end attributes row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets in the x-axis.

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      The matrix representation of the image :
      open square brackets table row 1 0 row 0 cell negative 1 end cell end table close square brackets open square brackets table row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets equals open square brackets table row cell 1 left parenthesis 2 right parenthesis plus 0 left parenthesis negative 2 right parenthesis end cell cell 1 left parenthesis 6 right parenthesis plus 0 left parenthesis negative 3 right parenthesis end cell cell 1 left parenthesis 5 right parenthesis plus 0 left parenthesis negative 8 right parenthesis end cell cell 1 left parenthesis 2 right parenthesis plus 0 left parenthesis negative 5 right parenthesis end cell row cell 0 left parenthesis 2 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 2 right parenthesis end cell cell 0 left parenthesis 6 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 3 right parenthesis end cell cell 0 left parenthesis 5 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 8 right parenthesis end cell cell 0 left parenthesis 2 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis negative 5 right parenthesis end cell end table close square brackets equals open square brackets table attributes columnalign center center right left end attributes row 2 6 5 2 row cell negative 2 end cell cell negative 3 end cell cell negative 8 end cell cell negative 5 end cell end table close square brackets

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      Since, the point lies on the x-axis. Then, the intersection of the line A'B and the x-axis gives the required point

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      Since, the point lies on the x-axis. Then, the intersection of the line A'B and the x-axis gives the required point.

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