Question

Part D

Answer 1

The shrink or stretch of the image of PQRS which is the dilation of PQRS is given as follows;

• The rule of the dilation is $\underline{D_{\frac{1}{2} }}$, followed by a reflection across the x-axis

Reason:

The given completed  table is presented as follows;

$\begin{array}{|cc|c|} \mathbf{Original \ Coordinates}&&\mathbf{New \ Coordinates}\\(x, \ y)&&(,)\\K(3, \, 2)&&P(6, \, -4)\\L(1, \, 1)&&Q(2, \, -2)\\M(1, \, 3)&&R(2, \, -6)\\N(3, \, 3)&&S(6, \, -6)\end{array}$

Length of the side $\overline{LM}$ = 3 - 1 = 2

Length of the side $\overline{MN}$ = 3 - 1 = 2

Length of the side $\overline{NK}$ = 3 - 2 = 1

Length of the side $\overline{LK}$ = $\sqrt{(2 - 1)^2 + (3 - 1)^2} = \sqrt{5}$

Length of the side $\overline{RQ}$ = -2 - (-6) = 4

Length of the side $\overline{RS}$ = 6 - 2 = 4

Length of the side $\overline{SP}$ = -4 - (-6) = 2

Length of the side $\overline{QP}$ = $\sqrt{(6 - 2)^2 + (-4 - (-2))^2} = 2\cdot \sqrt{5}$

$\dfrac{\overline{LM}}{\overline{RQ}} = \dfrac{\overline{MN}}{\overline{RS}} = \dfrac{\overline{NK}}{\overline{SP}} = \dfrac{\overline{LK}}{\overline{QP}} = \dfrac{1}{2}$

However, the coordinates of the image of PQRS, which is P'Q'R'S', following a dilation are'

$P' \left(\dfrac{1}{2} \times 6, \, \dfrac{1}{2} \times (-4) \right) = P'(3, \, -2)$

$Q' \left(\dfrac{1}{2} \times 2, \, \dfrac{1}{2} \times (-2) \right) = Q'(1, \, -1)$

$R' \left(\dfrac{1}{2} \times 2, \, \dfrac{1}{2} \times (-6) \right) = R'(1, \, -3)$

$S' \left(\dfrac{1}{2} \times 6, \, \dfrac{1}{2} \times (-6) \right) = S'(3, \, -3)$  

The transformation of the coordinates of P'Q'R'S', to KLMN are;

• (x, y) → (x, -y)

which is equivalent to a reflection across the x-axis

Therefore, the transformation that gives the coordinate of KLMN from PQRS is a dilation by a scale factor of $\underline{\dfrac{1}{2}}$, which is a rule of dilation of $\underline{D_{\frac{1}{2} }}$, followed by a reflection across the x-axis

Learn more here:

brainly.com/question/12382913