I see that you've uploaded a photo of a geometry problem related to dilations. Let's go over the problems together.

Problem 1:

"Segment AB measures 3 centimeters. Point O is the center of dilation. How long is the image of AB after a dilation with a scale factor of $\frac{1}{5}$?"

To solve:

• The length of a segment after dilation is given by multiplying the original length by the scale factor.
• Original length $AB = 3$ cm.
• Scale factor $\frac{1}{5}$.

So, the new length of segment $AB' = 3 \times \frac{1}{5} = 0.6$ cm.

Problem 2:

"How long is the image of AB after a dilation with a scale factor of $s$?"

To solve:

• Again, use the formula for dilation: the new length is the original length multiplied by the scale factor.
• Original length $AB = 3$ cm.
• Scale factor $s$.

So, the new length of segment $AB' = 3 \times s$ cm.

Problem 3:

"Here are points A and B. Point F is the image of point B using point A as the center of dilation and a scale factor of $\frac{1}{3}$. Plot point F."

• Point A is the center of dilation.
• The image of B (point F) will be located at one-third of the distance from A to B, since the scale factor is $\frac{1}{3}$.

To find F:

• Measure the distance between A and B on the graph.
• Multiply this distance by $\frac{1}{3}$ and locate point F along the line joining A and B, but only one-third of the way.

Problem 4:

"Triangle ABC was transformed into triangle DFE using a dilation. Label the center of dilation P. Then determine the scale factor."

• Look at the transformation of triangle ABC to triangle DFE.
• Identify a common point of dilation by tracing the lines through corresponding vertices (like A and D, B and E, C and F). The point where these lines intersect is the center of dilation, P.
• To determine the scale factor, divide the length of a side of triangle DFE by the corresponding side of triangle ABC.

Problems 5 and 6:

For these problems, you are asked to draw the dilated image of rectangle PQRS.

• Problem 5: Using point R as the center and a scale factor of 2, you will double the distance of each vertex from R and redraw the rectangle accordingly.
• Problem 6: Using point O as the center and a scale factor of $\frac{1}{2}$, you will halve the distance of each vertex from O.

Do you need help plotting points for these problems or with more details?

Here are five related questions to explore further:

1. What happens to the shape of a figure when the scale factor is greater than 1?
2. What is the effect of a negative scale factor in dilation?
3. How do you determine the center of dilation when both the original figure and its image are given?
4. How can you apply dilation to real-world problems, like enlarging a blueprint?
5. What is the relationship between dilation and similarity in geometry?

Tip: When performing dilations, always keep in mind the center of dilation, as it affects the direction and distance of the transformation!