Question

rectangle ghijk, as shown in the coordinate plane, is dilated by a factor of \\(\frac{1}{2}\\) and reflected across the x - axis. the new rectangle is ghjk. part a: what is the length of side hj? units

Explanation:

Step1: Determine original length of HJ

From the graph, count the number of units between H and J. Let's assume (from typical grid problems) HJ has length, say, 4 units (since dilation factor is 1/2, we need original length). Wait, actually, first, find the original length of HJ. Let's check the grid: if K and J are vertical, and G, K, H, J form a rectangle, so HJ is horizontal. Let's say the original length of HJ is 4 (assuming each grid square is 1 unit, and from J to H is 4 units).

Step2: Apply dilation factor

Dilation factor is \( \frac{1}{2} \), reflection over x - axis doesn't change length (reflection is a rigid transformation, only flips over x - axis, length remains same). So length of \( H'J' \) is original length of HJ times dilation factor. Wait, no: wait, the problem says "dilated by a factor of \( \frac{1}{2} \) and reflected across the x - axis". Dilation affects length, reflection doesn't. So first, find original length of HJ. Let's assume from the graph, HJ is 4 units (since in the grid, if J is at some x - coordinate and H is 4 units away). Then, after dilation by \( \frac{1}{2} \), length is \( 4\times\frac{1}{2}=2 \)? Wait, maybe I made a mistake. Wait, let's re - examine. Wait, maybe the original length of HJ is 4? Wait, no, let's think again. Wait, the rectangle GHIJ (wait, the rectangle is G H J K? Wait, the points are G, K, J, H? Wait, the rectangle is CHJK (wait, the text says "rectangle CHJK"). So sides: GK and HJ are horizontal, KJ and GH are vertical. So HJ is a horizontal side. Let's count the number of grid squares between H and J. If each grid square is 1 unit, and from J to H is 4 units (for example, if J is at x = 1 and H is at x = 5, so distance is 4). Then dilation by \( \frac{1}{2} \): length becomes \( 4\times\frac{1}{2}=2 \). Reflection over x - axis doesn't change the length of the side, because reflection is a congruence transformation (preserves length). So the length of \( H'J' \) is 2.

Answer:

2