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Quadrilateral ABCD with vertices A(0, 6), B(-3, -6), C(-9, -6), and D(-12, -3): a) dilation with scale factor of 1/3 centered at the origin b) translation along the vector <-5,-1>

Sagot :

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a) The points of the new quadrillateral are [tex]A'(x,y) = (0, 2)[/tex], [tex]B'(x,y) = (-1, -2)[/tex], [tex]C'(x,y) = \left(-3,-2\right)[/tex] and [tex]D'(x,y) = (-4, -1)[/tex], respectively.

b) The points of the new quadrillateral are [tex]A'(x,y) = (-5, 5)[/tex], [tex]B'(x,y) = (-8,-7)[/tex], [tex]C'(x,y) = (-13, -7)[/tex] and [tex]D'(x,y) = (-17, -4)[/tex], respectively.

How to perform transformations with points

a) A dillation centered at the origin is defined by following operation:

[tex]P'(x,y) = k\cdot P(x,y)[/tex] (1)

Where:

  • [tex]P(x,y)[/tex] - Original point
  • [tex]P'(x,y)[/tex] - Dilated point.

If we know that [tex]k = \frac{1}{3}[/tex], [tex]A(x,y) = (0,6)[/tex], [tex]B(x,y) = (-3,-6)[/tex], [tex]C(x,y) = (-9, -6)[/tex] and [tex]D(x,y) = (-12, -3)[/tex], then the new points of the quadrilateral are:

[tex]A'(x,y) = \frac{1}{3}\cdot (0,6)[/tex]

[tex]A'(x,y) = (0, 2)[/tex]

[tex]B'(x,y) = \frac{1}{3} \cdot (-3,-6)[/tex]

[tex]B'(x,y) = (-1, -2)[/tex]

[tex]C'(x,y) = \frac{1}{3}\cdot (-9,-6)[/tex]

[tex]C'(x,y) = \left(-3,-2\right)[/tex]

[tex]D'(x,y) = \frac{1}{3}\cdot (-12,-3)[/tex]

[tex]D'(x,y) = (-4, -1)[/tex]

The points of the new quadrillateral are [tex]A'(x,y) = (0, 2)[/tex], [tex]B'(x,y) = (-1, -2)[/tex], [tex]C'(x,y) = \left(-3,-2\right)[/tex] and [tex]D'(x,y) = (-4, -1)[/tex], respectively. [tex]\blacksquare[/tex]

b) A translation along a vector is defined by following operation:

[tex]P'(x,y) = P(x,y) +T(x,y)[/tex] (2)

Where [tex]T(x,y)[/tex] is the transformation vector.

If we know that [tex]T(x,y) = (-5,-1)[/tex], [tex]A(x,y) = (0,6)[/tex], [tex]B(x,y) = (-3,-6)[/tex], [tex]C(x,y) = (-9, -6)[/tex] and [tex]D(x,y) = (-12, -3)[/tex],

[tex]A'(x,y) = (0,6) + (-5, -1)[/tex]

[tex]A'(x,y) = (-5, 5)[/tex]

[tex]B'(x,y) = (-3, -6) + (-5, -1)[/tex]

[tex]B'(x,y) = (-8,-7)[/tex]

[tex]C'(x,y) = (-9, -6) + (-5, -1)[/tex]

[tex]C'(x,y) = (-13, -7)[/tex]

[tex]D'(x,y) = (-12,-3)+(-5,-1)[/tex]

[tex]D'(x,y) = (-17, -4)[/tex]

The points of the new quadrillateral are [tex]A'(x,y) = (-5, 5)[/tex], [tex]B'(x,y) = (-8,-7)[/tex], [tex]C'(x,y) = (-13, -7)[/tex] and [tex]D'(x,y) = (-17, -4)[/tex], respectively. [tex]\blacksquare[/tex]

To learn more on transformation rules, we kindly invite to check this verified question: https://brainly.com/question/4801277

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