Question

quadrilateral fghi is dilated by a scale factor of \\(\frac{2}{3}\\) to form quadrilateral fghi. what is the measure of side if? (image of quadrilateral fghi with sides: if = 18, fg = 9, gh = 27, hi = 18.7)

Explanation:

Step1: Identify the original side length

In quadrilateral FGHI, the length of side IF is 18 (from the diagram: vertical side from I to F is 18).

Step2: Apply the dilation scale factor

Dilation with a scale factor of $\frac{2}{3}$ means the new side length (I'F') is the original length multiplied by the scale factor. Wait, no—wait, actually, when a figure is dilated, the corresponding sides of the image (F'G'H'I') are related to the original (FGHI) by the scale factor. Wait, but the question is about side I'F'? Wait, no, the original side IF is 18. Wait, let's check the diagram again. Wait, the original quadrilateral: F to G is 9, F to I is 18 (vertical side). So when we dilate by scale factor $\frac{2}{3}$, the length of I'F' (the corresponding side in the dilated quadrilateral) is original length (IF) times scale factor? Wait, no—wait, dilation: if the scale factor is $\frac{2}{3}$, then the image side length is original length scale factor? Wait, no, wait: dilation scale factor $k$ means the image length is $k$ times the original length. Wait, but let's confirm the original side IF: from the diagram, IF is 18 (vertical side from I to F). So to find I'F', we multiply 18 by $\frac{2}{3}$? Wait, no, wait—wait, maybe I misread. Wait, the original quadrilateral: F to G is 9, F to I is 18 (vertical), G to H is 27 (vertical), I to H is 12.7 (slanted). Wait, when we dilate by scale factor $\frac{2}{3}$, the corresponding side I'F' (in the dilated quadrilateral F'G'H'I') will be the original IF length multiplied by the scale factor? Wait, no—wait, dilation: if the scale factor is $\frac{2}{3}$, then the image is smaller (since $\frac{2}{3} < 1$). So original IF is 18, so I'F' = 18 $\frac{2}{3}$? Wait, 18 $\frac{2}{3}$ = 12? Wait, no, wait—wait, maybe I got the direction wrong. Wait, the problem says "dilated by a scale factor of $\frac{2}{3}$ to form quadrilateral F'G'H'I'". So the original is FGHI, image is F'G'H'I'. So corresponding sides: I'F' corresponds to IF. So length of I'F' = length of IF scale factor. IF is 18, so 18 $\frac{2}{3}$ = 12? Wait, but let's check again. Wait, maybe the original side IF is 18, so multiplying by $\frac{2}{3}$: 18 (2/3) = 12. Wait, but let's confirm the diagram. The vertical side from F to I is labeled 18, yes. So the dilated side I'F' should be 18 (2/3) = 12. Wait, but maybe I made a mistake. Wait, no—dilation scale factor: if k is the scale factor, then image length = k original length. So here k = 2/3, original IF = 18, so I'F' = 18 * (2/3) = 12.

Answer:

12