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XYZ is a triangle with vertices X(1,-3), Y(7,5), Z(-3,5). L is the midpoint of side XZ. If O is the origin, express XY, YZ, ZX as column vectors.

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Final answer:

XY, YZ, and ZX can be represented as the column vectors (6, 8), (-10, 0), and (4, -8) respectively, derived by subtracting the coordinates of the initial points from the terminal points.

Step-by-step explanation:

To express XY, YZ, and ZX as column vectors, we first define them in a two-dimensional coordinate system (x, y) given that this problem involves only x and y coordinates and no z-axis is mentioned. We will find the vectors by subtracting the coordinates of the initial point from the coordinates of the terminal point.

For vector XY, with X(1, -3) as the starting point and Y(7, 5) as the ending point:

  • Ax = x-coordinates of Y minus x-coordinates of X = 7 - 1 = 6
  • Ay = y-coordinates of Y minus y-coordinates of X = 5 - (-3) = 8

Hence, the vector XY is:

\[ \begin{pmatrix} 6 \\ 8 \end{pmatrix} \]

For vector YZ, with Y(7, 5) as the starting point and Z(-3, 5) as the ending point:

  • Ax = x-coordinates of Z minus x-coordinates of Y = -3 - 7 = -10
  • Ay = y-coordinates of Z minus y-coordinates of Y = 5 - 5 = 0

The vector YZ is:

\[ \begin{pmatrix} -10 \\ 0 \end{pmatrix} \]

Finally, for vector ZX, with Z(-3, 5) as the starting point and X(1, -3) as the ending point:

  • Ax = x-coordinates of X minus x-coordinates of Z = 1 - (-3) = 4
  • Ay = y-coordinates of X minus y-coordinates of Z = -3 - 5 = -8

The vector ZX is:

\[ \begin{pmatrix} 4 \\ -8 \end{pmatrix} \]

Each of these calculations takes into account the analytical relationships between vectors and their x- and y-components, which together form right triangles as described in the references provided.

User Joel Cornett
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