Question

1. each triangle is dilated using the origin as the center of dilation to create a new triangle. determine the scale factor and write the rule applied to create the new triangle.

a. c(-3, 4) a(5,1) t(-2, -5)
c(-6, 8) a(10, 2) t(-4, -10)
b. x(9,6) y(3,9) z(6,3)
x(3,2) y(1,3) z(2,1)
prepare
solve each equation for x.

1. x + 105 = 180
2. 2x + 65 = 180
3. 45 + 4x - 15 = 180
4. 90 + 2x = 180

Explanation:

2a.

Step1: Find scale - factor for x - coordinates

For point C, $\frac{-6}{-3}=2$, for point A, $\frac{10}{5}=2$, for point T, $\frac{-4}{-2}=2$.

Step2: Find scale - factor for y - coordinates

For point C, $\frac{8}{4}=2$, for point A, $\frac{2}{1}=2$, for point T, $\frac{-10}{-5}=2$.

Step3: Write the rule

The rule for dilation with a scale factor $k$ centered at the origin is $(x,y)\to(kx,ky)$. Here $k = 2$, so the rule is $(x,y)\to(2x,2y)$.

Step1: Find scale - factor for x - coordinates

For point X, $\frac{3}{9}=\frac{1}{3}$, for point Y, $\frac{1}{3}=\frac{1}{3}$, for point Z, $\frac{2}{6}=\frac{1}{3}$.

Step2: Find scale - factor for y - coordinates

For point X, $\frac{2}{6}=\frac{1}{3}$, for point Y, $\frac{3}{9}=\frac{1}{3}$, for point Z, $\frac{1}{3}=\frac{1}{3}$.

Step3: Write the rule

The rule for dilation with a scale factor $k$ centered at the origin is $(x,y)\to(kx,ky)$. Here $k=\frac{1}{3}$, so the rule is $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$.

Step1: Isolate x

Subtract 105 from both sides of the equation $x + 105=180$. So $x=180 - 105$.

Answer:

Scale factor: $2$, Rule: $(x,y)\to(2x,2y)$

2b.