Question

triangle stv was dilated with the origin as the center of dilation to form δ stv. what is the scale factor of the dilation? \\(\frac{1}{3}\\) \\(\frac{2}{3}\\) \\(\frac{3}{2}\\) \\(\frac{3}{1}\\)

Explanation:

Step1: Identify Coordinates

Find coordinates of a vertex (e.g., S and S'). Let's assume S is at (0, 6) and S' is at (0, 2) (from the grid, estimating heights).

Step2: Calculate Scale Factor

Scale factor \( k = \frac{\text{Length of image}}{\text{Length of original}} \). For S to S': \( k = \frac{2}{6} = \frac{1}{3} \)? Wait, no—wait, maybe I mixed up. Wait, dilation: if S' is the image, then scale factor is \( \frac{\text{image length}}{\text{original length}} \). Wait, maybe S is (0, 6) and S' is (0, 2)? No, wait the grid: Let's check the base. Original base TV: from x=-3 to x=3 (length 6), image T'V': from x=-1 to x=1 (length 2). So scale factor \( k = \frac{2}{6} = \frac{1}{3} \)? Wait no, wait the options: 1/3, 2/3, 3/2, 3/1. Wait maybe I got original and image reversed. Wait, dilation from STV to S'TV', so S'TV' is the image. Wait, maybe S is (0, 6) and S' is (0, 2)? No, wait the y-axis: S is at (0, 6) (since from origin up 6), S' is at (0, 2)? No, wait the grid lines: each square is 1 unit. Let's count the y-coordinate of S: from origin (0,0) up to S: let's see, the original triangle's height (from S to TV) is 6 units (from y=6 to y=0? No, TV is at y=-6? Wait no, the graph: T and V are at the bottom, S at the top. Let's take S: (0, 6), S': (0, 2). Then scale factor is \( \frac{2}{6} = \frac{1}{3} \)? Wait but maybe I made a mistake. Wait, another way: scale factor is \( \frac{\text{image length}}{\text{original length}} \). If S'TV' is the image, then length of S'T' (or S'V') vs ST (SV). Wait, maybe the original height is 6, image height is 2, so scale factor 2/6 = 1/3? But wait the options have 1/3. Wait, let's check the base: original TV: from x=-3 to x=3 (length 6), image T'V': from x=-1 to x=1 (length 2). So 2/6 = 1/3. So scale factor is 1/3.

Answer:

\(\frac{1}{3}\) (Option: \(\boldsymbol{\frac{1}{3}}\))