Question

exercises perform the dilation. 1. k = 1/3 (-12, 3) (6, 33) 2. k = 4 (-12, 3) (2, -3) 3. k = 2/7 (-21, 49) (2, 35) 4. k = 2 (-1, 3) (2, 3) 5. k = 2/3 (27, 9) (-6, 36) 6. k = 4/9 (-18, 27) (27, 45) 1. k = 3 (-1, 13) (12, 13) 2. k = 3/2 (-2, 4) (-12, 6)

Explanation:

1. For problem 1 ($k = \frac{1}{3}$, points $(- 12,3)$ and $(6,33)$):

• Explanation of dilation formula:
• The formula for dilation of a point $(x,y)$ with a scale - factor $k$ centered at the origin is $(kx,ky)$.
• Dilation of $(-12,3)$:
• Multiply each coordinate by $k=\frac{1}{3}$.
• For the $x$ - coordinate: $x_1=\frac{1}{3}\times(-12)=-4$.
• For the $y$ - coordinate: $y_1=\frac{1}{3}\times3 = 1$.
• Dilation of $(6,33)$:
• Multiply each coordinate by $k = \frac{1}{3}$.
• For the $x$ - coordinate: $x_2=\frac{1}{3}\times6 = 2$.
• For the $y$ - coordinate: $y_2=\frac{1}{3}\times33 = 11$.
• The dilated points are $(-4,1)$ and $(2,11)$.

1. For problem 2 ($k = 4$, points $(-12,3)$ and $(2, - 3)$):

• Dilation of $(-12,3)$:
• Multiply each coordinate by $k = 4$.
• For the $x$ - coordinate: $x_1=4\times(-12)=-48$.
• For the $y$ - coordinate: $y_1=4\times3 = 12$.
• Dilation of $(2,-3)$:
• Multiply each coordinate by $k = 4$.
• For the $x$ - coordinate: $x_2=4\times2 = 8$.
• For the $y$ - coordinate: $y_2=4\times(-3)=-12$.
• The dilated points are $(-48,12)$ and $(8,-12)$.

1. For problem 3 ($k=\frac{2}{7}$, points $(-21,49)$ and $(2,35)$):

• Dilation of $(-21,49)$:
• Multiply each coordinate by $k=\frac{2}{7}$.
• For the $x$ - coordinate: $x_1=\frac{2}{7}\times(-21)=-6$.
• For the $y$ - coordinate: $y_1=\frac{2}{7}\times49 = 14$.
• Dilation of $(2,35)$:
• Multiply each coordinate by $k=\frac{2}{7}$.
• For the $x$ - coordinate: $x_2=\frac{2}{7}\times2=\frac{4}{7}$.
• For the $y$ - coordinate: $y_2=\frac{2}{7}\times35 = 10$.
• The dilated points are $(-6,14)$ and $(\frac{4}{7},10)$.

1. For problem 4 ($k = 2$, points $(-1,3)$ and $(2,3)$):

• Dilation of $(-1,3)$:
• Multiply each coordinate by $k = 2$.
• For the $x$ - coordinate: $x_1=2\times(-1)=-2$.
• For the $y$ - coordinate: $y_1=2\times3 = 6$.
• Dilation of $(2,3)$:
• Multiply each coordinate by $k = 2$.
• For the $x$ - coordinate: $x_2=2\times2 = 4$.
• For the $y$ - coordinate: $y_2=2\times3 = 6$.
• The dilated points are $(-2,6)$ and $(4,6)$.

1. For problem 5 ($k=\frac{2}{3}$, points $(27,9)$ and $(-6,36)$):

• Dilation of $(27,9)$:
• Multiply each coordinate by $k=\frac{2}{3}$.
• For the $x$ - coordinate: $x_1=\frac{2}{3}\times27 = 18$.
• For the $y$ - coordinate: $y_1=\frac{2}{3}\times9 = 6$.
• Dilation of $(-6,36)$:
• Multiply each coordinate by $k=\frac{2}{3}$.
• For the $x$ - coordinate: $x_2=\frac{2}{3}\times(-6)=-4$.
• For the $y$ - coordinate: $y_2=\frac{2}{3}\times36 = 24$.
• The dilated points are $(18,6)$ and $(-4,24)$.

1. For problem 6 ($k=\frac{4}{9}$, points $(-18,27)$ and $(27,45)$):

• Dilation of $(-18,27)$:
• Multiply each coordinate by $k=\frac{4}{9}$.
• For the $x$ - coordinate: $x_1=\frac{4}{9}\times(-18)=-8$.
• For the $y$ - coordinate: $y_1=\frac{4}{9}\times27 = 12$.
• Dilation of $(27,45)$:
• Multiply each coordinate by $k=\frac{4}{9}$.
• For the $x$ - coordinate: $x_2=\frac{4}{9}\times27 = 12$.
• For the $y$ - coordinate: $y_2=\frac{4}{9}\times45 = 20$.
• The dilated points are $(-8,12)$ and $(12,20)$.

Answer:

1. $(-4,1),(2,11)$
2. $(-48,12),(8,-12)$
3. $(-6,14),(\frac{4}{7},10)$
4. $(-2,6),(4,6)$
5. $(18,6),(-4,24)$
6. $(-8,12),(12,20)$