Question

the trapezoid fghi is a dilation of the trapezoid fghi. 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 Corresponding Points

Find coordinates of a point in original trapezoid (FGHI) and its image in dilated trapezoid (F'G'H'I'). Let's take point F and F':

• Coordinates of F: $(3, 0)$ (Wait, looking at graph, F is at (3,0)? Wait no, let's check again. Wait F is at (3, 0)? Wait no, looking at the grid, F is at (3, 0)? Wait no, F is at (3, 0)? Wait actually, looking at the graph, F is at (3, 0)? Wait no, let's take G and G':
• G: Let's see, G is at (3, 3)? Wait no, original trapezoid FGHI: G is at (3, 3)? Wait no, looking at the blue points: G is at (3, 3)? Wait no, H is at (-3, 2), I is at (-3, -3), F is at (3, 0), G is at (3, 3)? Wait no, maybe better to take vertical side. Let's take FG: from F(3,0) to G(3,3), so length is 3 - 0 = 3? Wait no, y-coordinate: G is at (3, 3), F is at (3, 0), so length FG is 3 - 0 = 3. Then F'G': F' is at (9, 0), G' is at (9, 9), so length F'G' is 9 - 0 = 9? Wait no, G' is at (9, 9)? Wait the graph: F' is at (9, 0) (since x=9, y=0), G' is at (9, 9) (x=9, y=9). So length F'G' is 9 - 0 = 9? Wait no, vertical distance: from y=0 to y=9, so length 9. Original FG: F is at (3, 0), G is at (3, 3), so vertical distance 3 - 0 = 3. So scale factor is length of image / length of original = 9 / 3 = 3? Wait no, wait maybe I messed up coordinates. Wait let's check H and H': H is at (-3, 2), H' is at (-9, 6). So x-coordinate: -3 to -9, difference is -6, but length is absolute. Wait H: (-3, 2), H': (-9, 6). The vertical distance for H to I: H is at (-3, 2), I is at (-3, -3), so vertical length is 2 - (-3) = 5? Wait no, 2 - (-3) = 5? Wait 2 - (-3) = 5? Wait 2 + 3 = 5. Then H' to I': H' is at (-9, 6), I' is at (-9, -9), so vertical length is 6 - (-9) = 15. Then scale factor is 15 / 5 = 3. Alternatively, take horizontal distance? Wait no, dilation scale factor is ratio of corresponding side lengths. Let's take FG: original FG: from F(3,0) to G(3,3) (wait no, F is at (3,0), G is at (3,3)? Wait no, looking at the blue points: F is at (3,0), G is at (3,3), H is at (-3,2), I is at (-3,-3). So FG is vertical from (3,0) to (3,3), length 3. F'G' is vertical from (9,0) to (9,9), length 9. So scale factor is 9/3 = 3. Wait but let's check another side: HI: from H(-3,2) to I(-3,-3), length is 2 - (-3) = 5. H'I': from H'(-9,6) to I'(-9,-9), length is 6 - (-9) = 15. 15/5 = 3. So scale factor is 3. Wait but maybe I made a mistake in coordinates. Wait let's check x-coordinates: original H is at x=-3, H' at x=-9. -9 / -3 = 3. Original F is at x=3, F' at x=9. 9/3=3. So scale factor is 3. Wait but let's confirm with another point. I is at x=-3, I' at x=-9. -9 / -3 = 3. So yes, scale factor is 3.

Wait maybe my initial coordinate reading was wrong. Let's re-express:

Original trapezoid FGHI:

• F: (3, 0)
• G: (3, 3)
• H: (-3, 2)
• I: (-3, -3)

Dilated trapezoid F'G'H'I':

• F': (9, 0)
• G': (9, 9)
• H': (-9, 6)
• I': (-9, -9)

Now, let's take the length of FG (vertical side): from F(3,0) to G(3,3), length is 3 - 0 = 3 (vertical distance).

Length of F'G' (vertical side): from F'(9,0) to G'(9,9), length is 9 - 0 = 9.

Scale factor = length of image / length of original = 9 / 3 = 3.

Alternatively, take the horizontal distance from F to H: F is at x=3, H at x=-3, distance is 3 - (-3) = 6.

F' to H': F' at x=9, H' at x=-9, distance is 9 - (-9) = 18. 18 / 6 = 3.

So scale factor is 3.

Step2: Confirm with Another Side

Take HI: from H(-3,2) to I(-3,-3), length is 2 - (-3) = 5 (vertical distance).

H'I': from H'(-9,6) to I'(-9,-9), length is 6 - (-9) = 15. 15 / 5 = 3.

Consistent, so scale factor is 3.

Answer:

3