Question

the trapezoid pqrs is a dilation of the trapezoid pqrs. what is the scale factor of the dilation?
simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Identify coordinates of a point and its image

Let's take point \( P \) and its image \( P' \). From the graph, \( P \) has coordinates \( (3, 4) \) (wait, actually looking at the grid, let's check again. Wait, \( P \) is at \( (3, 4) \)? Wait no, looking at the orange trapezoid: \( P \) is at \( (3, 4) \)? Wait, maybe better to take \( P \) and \( P' \). Let's see: \( P \) is at \( (3, 4) \)? Wait, no, let's check the coordinates. Let's take \( P \): from the grid, \( P \) is at \( (3, 4) \)? Wait, \( P' \) is at \( (6, 8) \). Wait, maybe \( P \) is at \( (3, 4) \) and \( P' \) is at \( (6, 8) \). Wait, or maybe \( P \) is at \( (3, 4) \), \( P' \) at \( (6, 8) \). Let's confirm. Alternatively, take \( Q \) and \( Q' \). \( Q \) is at \( (3, 5) \)? Wait, no, looking at the orange trapezoid: \( Q \) is at \( (3, 5) \)? Wait, \( Q' \) is at \( (6, 10) \). Wait, let's check the coordinates properly. Let's list the coordinates:

For trapezoid \( PQRS \) (orange):

• \( P \): Let's see, the x-coordinate: from the grid, \( P \) is at x=3, y=4. So \( P(3, 4) \)
• \( P' \) (blue) is at x=6, y=8. So \( P'(6, 8) \)

Step2: Calculate the scale factor

The scale factor \( k \) of a dilation is given by the ratio of the coordinates of the image to the original. So for the x-coordinate: \( \frac{6}{3} = 2 \), for the y-coordinate: \( \frac{8}{4} = 2 \). Let's check another point. Take \( S \) and \( S' \). \( S \) is at \( (5, -1) \)? Wait, no, \( S \) is at \( (5, -1) \)? Wait, \( S' \) is at \( (10, -2) \). So \( \frac{10}{5} = 2 \), \( \frac{-2}{-1} = 2 \). So the scale factor is 2.

Wait, let's confirm with \( Q \) and \( Q' \). \( Q \) is at \( (3, 5) \), \( Q' \) is at \( (6, 10) \). \( \frac{6}{3} = 2 \), \( \frac{10}{5} = 2 \). Yep, that works. So the scale factor is the ratio of the image coordinates to the original coordinates, which is \( \frac{\text{coordinate of } P'}{\text{coordinate of } P} = \frac{8}{4} = 2 \) (or \( \frac{6}{3} = 2 \)).

Answer:

2