Question

1. identify the scale factor used to graph the image below. k =

Explanation:

Step1: Determine original and image lengths

First, find the distance between \( H \) and another point (e.g., \( G \)) in the original figure, and the distance between \( H' \) and \( G' \) in the image. Let's use horizontal or vertical distances for simplicity. Looking at \( H \) and \( H' \): suppose \( H \) is at some x - coordinate, and \( H' \) is further. Let's count the grid units. If the original segment \( HH \) (wait, better: let's take the distance from \( H \) to \( G \) horizontally? Wait, maybe better to use the distance from \( H \) to \( H' \). Let's assume the original \( H \) to \( H \) (no, original triangle: let's find the length of \( HH \) in original and \( H'H' \) in image. Wait, actually, let's take the horizontal distance between \( H \) and, say, the x - coordinate. Suppose in the original, the distance from \( H \) to the vertical line through \( G \) is, say, 2 units, and in the image, it's 6 units? Wait, no, let's count the grid squares. Let's assume the original triangle has a base (from \( H \) to the other vertex) of length \( l \), and the image has length \( 3l \)? Wait, actually, let's look at the horizontal distance between \( H \) and \( H' \). If \( H \) is at x = a, and \( H' \) is at x = a + 8 (for example), and the original horizontal distance from \( H \) to the corresponding point in the original triangle is, say, \( \frac{8}{3} \)? No, wait, let's use the scale factor formula: scale factor \( k=\frac{\text{length of image segment}}{\text{length of original segment}} \).

Let's take the segment \( HH \) (no, \( H \) to \( H' \)): suppose in the original, the distance from \( H \) to the vertex \( I \) horizontally? Wait, maybe better to use the distance between \( H \) and \( G \) in the original and \( H' \) and \( G' \) in the image. Let's count the grid squares. Let's say in the original, the horizontal distance from \( H \) to \( G \) is 2 units, and in the image, it's 6 units. Then \( k = \frac{6}{2}=3 \)? Wait, no, let's do it properly. Let's assume the original triangle has a base (from \( H \) to the point directly below \( G \)) of length \( l \), and the image has length \( 3l \). Alternatively, let's take the distance from \( H \) to \( H' \). Suppose in the original, the distance between \( H \) and the x - coordinate of \( H \) (no, let's count the number of grid squares between \( H \) and \( H' \). If \( H \) is at x = 2 (for example) and \( H' \) is at x = 8, then the distance is 6, and the original distance from \( H \) to the corresponding point (before scaling) was 2, so \( k=\frac{6}{2} = 3 \). Wait, actually, looking at the graph, if we count the grid units: let's say the original triangle has a horizontal length (from \( H \) to the other vertex) of 2 units, and the image has 6 units, so scale factor \( k = 3 \). Wait, another way: the scale factor is the ratio of the lengths of corresponding sides. Let's take the vertical distance from \( G \) to \( I \) in the original and \( G' \) to \( I' \) in the image. If in the original, \( G \) is 2 units above \( I \), and in the image, \( G' \) is 6 units above \( I' \), then \( k=\frac{6}{2}=3 \).

Step2: Calculate the scale factor

Let's assume the original segment length (e.g., from \( H \) to \( H' \) in original is \( l \), and in image is \( 3l \)). So the scale factor \( k=\frac{\text{image length}}{\text{original length}} \). If we take the horizontal distance between \( H \) and \( H' \): suppose original length is \( x \), image length is \( 3x \), so \( k = 3 \).

Answer:

\( 3 \)