Question

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

Explanation:

Step1: Choose corresponding points

Let's take points \( U \) and \( U' \), or other corresponding vertices. Let's assume the original figure (larger) has a side length, and the image (smaller) has a corresponding side length. Let's count the grid units. Suppose from \( S \) to \( V \) in the original figure: let's say the horizontal distance from \( S \) to \( V \) is, for example, 10 units (counting grids), and from \( S' \) to \( V' \) is 5 units? Wait, no, let's check the vertical or horizontal. Wait, actually, the scale factor \( k \) is the ratio of the length of the image to the original. So if we take a side, say from \( T \) to \( U \) in the original and \( T' \) to \( U' \) in the image. Let's count the grid squares. Suppose the original segment \( TU \) spans, say, 2 units vertically and 2 units horizontally? Wait, maybe better to take a horizontal or vertical segment. Wait, looking at the x-axis: the original figure's point \( V \) is at some x-coordinate, and \( V' \) is halfway? Wait, actually, the scale factor is the ratio of the length of the image (smaller triangle) to the original (larger triangle). So if we measure the distance between two corresponding points. Let's take \( U \) and \( U' \). Suppose \( U \) is at (let's assign coordinates: let's say each grid is 1 unit. Let \( U \) be at (6, 10), \( U' \) at (4, 6)? Wait, no, maybe better to count the number of grid squares between corresponding points. Alternatively, notice that the image is half the size of the original. So the scale factor \( k \) is \( \frac{1}{2} \), because the image (smaller triangle) has sides that are half the length of the original (larger triangle). Let's verify: if the original triangle has a side length of, say, 4 units (in grid terms), the image has 2 units. So \( k = \frac{2}{4} = \frac{1}{2} \).

Step2: Confirm the scale factor

The scale factor is the ratio of the length of a side in the image (the transformed figure) to the length of the corresponding side in the original figure. By observing the grid, the image triangle is a dilation of the original triangle with a scale factor of \( \frac{1}{2} \), since each side of the image is half the length of the corresponding side in the original.

Answer:

\( \frac{1}{2} \)