Question

iii. mapping similar figures using dilations
topic practice
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.

1. y(15,6), z(20,10)

y(5, 2), z(6.7, 3.3)

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. 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. 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. which transformation(s) preserve congruence?

Explanation:

Step1: Recall scale - factor formula

For a dilation with the origin as the center, 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 $y
eq0$).

Step2: Solve for problem 2

For point $A(14,2)$ dilated to $A'(3.5,0.5)$:
The scale - factor $k=\frac{3.5}{14}=\frac{0.5}{2}=\frac{1}{4}$.
For point $B(18,6)$ dilated to $B'(4.5,1.5)$:
$k = \frac{4.5}{18}=\frac{1.5}{6}=\frac{1}{4}$.
For point $C(22,2)$ dilated to $C'(5.5,0.5)$:
$k=\frac{5.5}{22}=\frac{0.5}{2}=\frac{1}{4}$.
The rule for dilation is $(x,y)\to(\frac{1}{4}x,\frac{1}{4}y)$.

Step3: Solve for problem 4

For point $Q(2, - 3)$ dilated to $Q'(8,-12)$:
The scale - factor $k=\frac{8}{2}=\frac{-12}{-3}=4$.
For point $R(5,-3)$ dilated to $R'(20,-12)$:
$k=\frac{20}{5}=\frac{-12}{-3}=4$.
For point $S(5,-5)$ dilated to $S'(20,-20)$:
$k=\frac{20}{5}=\frac{-20}{-5}=4$.
For point $T(2,-5)$ dilated to $T'(8,-20)$:
$k=\frac{8}{2}=\frac{-20}{-5}=4$.
The rule for dilation is $(x,y)\to(4x,4y)$.

Step4: Solve for problem 6

For point $W(6,2)$ dilated to $W'(21,7)$:
The scale - factor $k=\frac{21}{6}=\frac{7}{2}$.
For point $X(12,2)$ dilated to $X'(42,7)$:
$k=\frac{42}{12}=\frac{7}{2}$.
For point $Y(12,4)$ dilated to $Y'(42,14)$:
$k=\frac{42}{12}=\frac{14}{4}=\frac{7}{2}$.
For point $Z(8,6)$ dilated to $Z'(28,21)$:
$k=\frac{28}{8}=\frac{21}{6}=\frac{7}{2}$.
The rule for dilation is $(x,y)\to(\frac{7}{2}x,\frac{7}{2}y)$.

Step5: Answer question 8

The transformations that preserve congruence are translations, rotations, and reflections. Dilations change the size of a figure and do not preserve congruence.

Answer:

1. Scale factor: $\frac{1}{4}$, Rule: $(x,y)\to(\frac{1}{4}x,\frac{1}{4}y)$
2. Scale factor: $4$, Rule: $(x,y)\to(4x,4y)$
3. Scale factor: $\frac{7}{2}$, Rule: $(x,y)\to(\frac{7}{2}x,\frac{7}{2}y)$
4. Translations, rotations, and reflections.