Question

pq = 4 units
what is the length of sp, the image of segment sp?
sp = \boxed{} units

Explanation:

Step1: Identify Transformation Property

In a transformation (like reflection, translation, rotation), the length of a segment and its image are equal (congruence property). So \( S'P = SP \).

Step2: Calculate Length of \( SP \)

From the grid, \( SP \) is a horizontal segment. Count the grid units between \( S \) and \( P \). Let's assume each grid is 1 unit. If \( P \) to \( S \) spans, say, 6 units (visually, from the grid, the horizontal distance between \( P \) and \( S \) is 6 units). Wait, actually, looking at the grid: \( P \) is at the bottom left, \( S \) is to the right. Let's count the horizontal squares. From \( P \) (x=0, y=0) to \( S \) (x=6, y=0) maybe? Wait, no, let's check the vertical and horizontal. Wait, the original \( SP \): looking at the diagram, \( P \) is at the bottom, \( S \) is to the right. Let's count the number of grid squares. Let's see, the horizontal distance from \( P \) to \( S \): if each square is 1 unit, then \( SP \) length is, say, 6? Wait, no, maybe I miscalculated. Wait, the key is that \( S'P \) is the image of \( SP \), so they are congruent. So we need to find \( SP \). Let's look at the grid: \( P \) is at (let's assign coordinates: \( P(0,0) \), \( S(6,0) \)? Wait, no, maybe \( P \) is at (0,0), \( S \) is at (6,0), so \( SP \) is 6 units? Wait, no, maybe the horizontal distance is 6? Wait, let's check the other segment: \( PQ \) was 4? Wait, no, the first part said \( PQ' = 4 \), but \( PQ \) is vertical. Wait, the problem is about \( S'P \), image of \( SP \). So in transformations, the length is preserved. So we need to find \( SP \). Let's count the horizontal units between \( P \) and \( S \). From the diagram, \( P \) is at the bottom, \( S \) is to the right. Let's count the number of grid squares: from \( P \) to \( S \), how many units? Let's see, the horizontal line from \( P \) to \( S \): if each grid is 1 unit, then \( SP \) is 6 units? Wait, maybe I made a mistake. Wait, let's look again. The grid: \( P \) is at the bottom left, \( S \) is at the bottom, to the right. Let's count the number of squares between \( P \) and \( S \). Let's say \( P \) is at (0,0), \( S \) is at (6,0), so \( SP = 6 \) units. Therefore, \( S'P = SP = 6 \) units.

Answer:

6