To graph the dilated image of quadrilateral $MNOP$ with a scale factor of $\frac{1}{3}$ and the origin as the center of dilation, follow these steps:

Step-by-step process:

1. Identify the coordinates of quadrilateral $MNOP$:

   • Suppose the vertices of quadrilateral $MNOP$ are $M(x_1, y_1)$, $N(x_2, y_2)$, $O(x_3, y_3)$, and $P(x_4, y_4)$.

2. Apply the scale factor:

   • The scale factor of $\frac{1}{3}$ means that each coordinate of the quadrilateral will be reduced to one-third of its original value.
   • The formula to calculate the new coordinates after dilation is: $(x', y') = \left( \frac{x}{3}, \frac{y}{3} \right)$
   • For each point $M, N, O, P$, apply this formula to get the new coordinates of the dilated image.

3. Example (Assume sample coordinates):

   • Let’s assume the original coordinates of $MNOP$ are: $M(6, 9), \quad N(3, 12), \quad O(9, 6), \quad P(12, 3)$
   • Now apply the scale factor $\frac{1}{3}$: $M' = \left( \frac{6}{3}, \frac{9}{3} \right) = (2, 3)$ $N' = \left( \frac{3}{3}, \frac{12}{3} \right) = (1, 4)$ $O' = \left( \frac{9}{3}, \frac{6}{3} \right) = (3, 2)$ $P' = \left( \frac{12}{3}, \frac{3}{3} \right) = (4, 1)$

4. Plot the original and dilated points:

   • First, plot the original quadrilateral $MNOP$ on a coordinate plane.
   • Then, plot the new dilated points $M'(2, 3)$, $N'(1, 4)$, $O'(3, 2)$, and $P'(4, 1)$.

5. Draw the dilated quadrilateral:

   • Connect the dilated points $M'N'O'P'$ to form the dilated image of the quadrilateral.

This gives you a smaller version of quadrilateral $MNOP$, with the size reduced by a factor of 3.

Would you like a visual example, or do you have specific coordinates you'd like to use?

Follow-up Questions:

1. What are the original coordinates of quadrilateral $MNOP$ in your problem?
2. Would you like help plotting the points on a coordinate plane?
3. How does the dilation affect the area of the quadrilateral?
4. What is the difference between dilation and other transformations like reflection or translation?
5. What happens if the scale factor is greater than 1?

Tip:

When using the origin as the center of dilation, the transformation affects both the size and position of the figure proportionally.