Question

iii. mapping similar figures using dilations
topic practice
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.

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

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

1. a(14, 2), b(18)

a(3.5, 0.5), b

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

e(2,0), f(2, -2.5), g(6, -2.5), h(5, -1)

1. q(2, -3), r

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

1. a(3, -4), b(4, -1), c(2, 2)

a(6, -8), b(8, -2), c(4, 4)

1. w(6, 2), x

w(21, 7)

1. do any of the dilation(s) in questions 1 - 6 preserve congruence? explain your reasoning.
2. which

presen

Explanation:

1. For the points \(X(10,10), Y(15,6), Z(20,10)\) and \(X'(3.3,3.3), Y'(5,2), Z'(6.7,3.3)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• For point \(X\) and \(X'\), if we consider the \(x\) - coordinates, the scale factor \(k_x=\frac{3.3}{10}\approx\frac{1}{3}\). For point \(Y\) and \(Y'\), \(k_x=\frac{5}{15}=\frac{1}{3}\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• For point \(X\) and \(X'\), considering the \(y\) - coordinates, \(k_y = \frac{3.3}{10}\approx\frac{1}{3}\). For point \(Y\) and \(Y'\), \(k_y=\frac{2}{6}=\frac{1}{3}\).
• The scale factor \(k = \frac{1}{3}\). The rule for dilation with the origin as the center is \((x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\).

1. For the points \(A(14,2)\) and \(A'(3.5,0.5)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• \(k_x=\frac{3.5}{14}=\frac{1}{4}\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• \(k_y=\frac{0.5}{2}=\frac{1}{4}\).
• The scale factor \(k=\frac{1}{4}\), and the rule is \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\).

1. For the points \(E(4,0), F(4, - 5), G(12, - 5), H(10, - 2)\) and \(E'(2,0), F'(2, - 2.5), G'(6, - 2.5), H'(5, - 1)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• For point \(E\) and \(E'\), \(k_x=\frac{2}{4}=\frac{1}{2}\). For point \(G\) and \(G'\), \(k_x=\frac{6}{12}=\frac{1}{2}\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• For point \(F\) and \(F'\), \(k_y=\frac{-2.5}{-5}=\frac{1}{2}\). For point \(H\) and \(H'\), \(k_y=\frac{-1}{-2}=\frac{1}{2}\).
• The scale factor \(k = \frac{1}{2}\), and the rule is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\).

1. For the points \(Q(2, - 3)\) and \(Q'(8, - 12)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• \(k_x=\frac{8}{2}=4\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• \(k_y=\frac{-12}{-3}=4\).
• The scale factor \(k = 4\), and the rule is \((x,y)\to(4x,4y)\).

1. For the points \(A(3, - 4), B(4, - 1), C(2,2)\) and \(A'(6, - 8), B'(8, - 2), C'(4,4)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• For point \(A\) and \(A'\), \(k_x=\frac{6}{3}=2\). For point \(B\) and \(B'\), \(k_x=\frac{8}{4}=2\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• For point \(A\) and \(A'\), \(k_y=\frac{-8}{-4}=2\). For point \(C\) and \(C'\), \(k_y=\frac{4}{2}=2\).
• The scale factor \(k = 2\), and the rule is \((x,y)\to(2x,2y)\).

1. For the points \(W(6,2)\) and \(W'(21,7)\):

• Step 1: Calculate the scale - factor for \(x\) - coordinates
• \(k_x=\frac{21}{6}=\frac{7}{2}\).
• Step 2: Calculate the scale - factor for \(y\) - coordinates
• \(k_y=\frac{7}{2}\).
• The scale factor \(k=\frac{7}{2}\), and the rule is \((x,y)\to(\frac{7}{2}x,\frac{7}{2}y)\).

1. For the question about congruence:

• Brief Explanation:
• Dilations preserve similarity. A dilation with a scale factor \(k

eq1\) changes the size of the figure. Congruent figures have the same size and shape. Since in all the dilations in questions 1 - 6, the scale factors are either greater than 1 or less than 1 (except \(k = 1\) which is not the case here), none of the dilations preserve congruence.

• Answer: No. Because the scale factors of all the dilations are not equal to 1, and dilations with non - unit scale factors change the size of the figures.

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: \(\frac{1}{2}\), Rule: \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)
4. Scale factor: \(4\), Rule: \((x,y)\to(4x,4y)\)
5. Scale factor: \(2\), Rule: \((x,y)\to(2x,2y)\)
6. Scale factor: \(\frac{7}{2}\), Rule: \((x,y)\to(\frac{7}{2}x,\frac{7}{2}y)\)
7. No. Because the scale factors of all the dilations are not equal to 1, and dilations with non - unit scale factors change the size of the figures.