Question

a. each figure is dilated using the origin as the center of dilation to create a new figure. determine the scale factor and write the rule applied to create the new figure.
x(10,10), y(15,6), z(20,10)
x(3.3, 3.3), y(5, 2), z(6.7, 3.3)
(x,y)→(\frac{1}{3}x,\frac{1}{3}y)

1. a(14, 2), b(18, 6), c(22,2)

a(3.5, 0.5), b(4.5, 1.5), c(5.5, 0.5)

1. d(0,0), f(4, -5), g(12, -5), h(10, -2)

d(0,0), f(2, -2.5), g(6, -2.5), h(5, -1)
(x,y)→(\frac{1}{2}x,\frac{1}{2}y)

1. q(2, -3), r(5, -3), s(5, -5), t(2, -5)

q(8, -12), r(20, -12), s(20, -20), t(8, -20)

1. a(2,1), b(4, -1), c(2, 2)

a(4,2), b(8, -2), c(4, 4)
(x,y)→(2x,2y)

1. w(6, 2), x(12, 2), y(12, 4), z(8, 6)

w(21, 7), x(42, 7), y(42, 14), z(28, 21)

1. do the dilation(s) in questions 1 - 6 preserve congruence? explain

the dilations do not preserve congruence because the scale factors are not equal

1. which transformation(s) preserve congruence? explain

Explanation:

Step1: Recall dilation formula

For a dilation with center at the origin, if a point $(x,y)$ is dilated to $(x',y')$, the scale - factor $k$ can be found by $k=\frac{x'}{x}=\frac{y'}{y}$ (assuming $x
eq0$ and $y
eq0$), and the rule is $(x,y)\to(kx,ky)$.

Step2: For the first set of points (e.g., for point $X(10,10)$ and $X'(3.3,3.3)$)

Calculate the scale - factor $k$. $k=\frac{3.3}{10}=0.33=\frac{1}{3}$. The rule is $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$.

Step3: For the second set of points (e.g., for point $A(14,2)$ and $A'(3.5,0.5)$)

Calculate the scale - factor. $k = \frac{3.5}{14}=\frac{1}{4}$ and $\frac{0.5}{2}=\frac{1}{4}$. The rule is $(x,y)\to(\frac{1}{4}x,\frac{1}{4}y)$.

Step4: For the third set of points (e.g., for point $Q(2, - 3)$ and $Q'(8,-12)$)

Calculate the scale - factor. $k=\frac{8}{2}=4$ and $\frac{-12}{-3}=4$. The rule is $(x,y)\to(4x,4y)$.

Step5: For the fourth set of points (e.g., for point $W(6,2)$ and $W'(21,7)$)

Calculate the scale - factor. $k=\frac{21}{6}=\frac{7}{2}$ and $\frac{7}{2}$. The rule is $(x,y)\to(\frac{7}{2}x,\frac{7}{2}y)$.

Step6: Answer question 8

Transformations that preserve congruence are translations, rotations, and reflections. Dilations change the size of a figure, so they do not preserve congruence. Translations (sliding a figure), rotations (turning a figure), and reflections (flipping a figure) only change the position or orientation of a figure but not its size or shape, thus preserving congruence.

Answer:

1. Scale factor: $\frac{1}{3}$, Rule: $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$
2. Scale factor: $\frac{1}{4}$, Rule: $(x,y)\to(\frac{1}{4}x,\frac{1}{4}y)$
3. Scale factor: $4$, Rule: $(x,y)\to(4x,4y)$
4. Scale factor: $\frac{7}{2}$, Rule: $(x,y)\to(\frac{7}{2}x,\frac{7}{2}y)$
5. Translations, rotations, and reflections preserve congruence because they do not change the size or shape of a figure; dilations change the size of a figure and thus do not preserve congruence.