Let’s delve into this problem! This worksheet requires performing dilations about the origin.

1. Dilating triangle ABC:

   • Original vertices: A, B, C.
   • Scale factor: $\frac{1}{2}$.
   • Multiply each coordinate by $\frac{1}{2}$:
     • $A'(-8, -12) \rightarrow (-8 \times \frac{1}{2}, -12 \times \frac{1}{2}) = (-4, -6)$
     • Similarly, calculate for B and C.

2. Dilating DEFG:

   • Original vertices: D, E, F, G.
   • Scale factor: $3$.
   • Multiply each coordinate by $3$:
     • Example: ( D'(-4 \rightarrow -12...Let me complete this step-by-step:

For triangle ABC (scale factor $\frac{1}{2}$):

• $A(-8, -12)$ → $A'(-4, -6)$,
• $B(2, 8)$ → $B'(1, 4)$,
• $C(-6, 6)$ → $C'(-3, 3)$.

For quadrilateral DEFG (scale factor $3$):

• $D(-1, 0)$ → $D'(-3, 0)$,
• $E(-3, 0)$ → $E'(-9, 0)$,
• $F(-1, -1)$ → $F'(-3, -3)$,
• $G(-2, 0)$ → $G'(-6, 0)$.

Would you like more detailed calculations or explanations?

Related Questions:

1. How does scaling by a fraction affect the shape's position?
2. How does scaling by a number greater than 1 affect the size?
3. Can dilation change the orientation of a shape?
4. What happens if the dilation center isn't the origin?
5. How is dilation related to similarity in geometry?

Tip:

Always double-check your new coordinates by multiplying each original coordinate by the scale factor!