Question

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

Explanation:

Step1: Find coordinates of S and S'

From the graph, S is at (0, 6) (assuming grid units, since S' is at (0, 2) and the triangle's height relates). Wait, actually, looking at the y - axis, S' is at (0, 2) and S is at (0, 6)? Wait, no, maybe better to check the length. Wait, the original triangle (STV) and the dilated (S'T'V'). Let's take the y - coordinate of S and S'. Let's say S is (0, 6) and S' is (0, 2). Wait, no, maybe the height. Alternatively, take the base. The base of the original triangle TV: from x=-3 to x = 3? Wait, no, looking at the grid, T is at (-3, -2) and V at (3, -2)? Wait, no, the dilated triangle S'T'V' has T' and V' closer. Wait, let's take the coordinates of S: let's assume S is (0, 6) (since the big triangle's top is at (0,6)), and S' is (0, 2) (the small triangle's top). Then the scale factor k is (coordinate of S')/(coordinate of S) = 2/6 = 1/3? Wait, no, wait, dilation from origin: the scale factor is the ratio of the image length to the original length. Let's take the y - coordinate. If S is (0, 6) and S' is (0, 2), then the scale factor k = 2/6 = 1/3? Wait, no, wait, maybe I got the direction wrong. Wait, the dilated triangle is smaller, so scale factor should be less than 1. Wait, the options are 1/3, 3/2, 3/1, 2/3. Wait, let's check the base. The original base TV: from x=-3 to x = 3, so length 6. The dilated base T'V': from x=-1 to x = 1, length 2. So scale factor is 2/6 = 1/3? Wait, no, wait, maybe the coordinates of S: let's look at the graph again. The big triangle (STV) has S at (0, 6) (since the y - axis, the big triangle's height is 6 units from y=-2 to y = 6? Wait, no, the small triangle S'T'V' has height from y=-2 to y = 2, so height 4? Wait, I think I made a mistake. Wait, let's take the coordinates of S: S is at (0, 6) (top of big triangle), S' is at (0, 2) (top of small triangle). Then the scale factor is (length of S')/(length of S) = 2/6 = 1/3. Wait, but let's confirm. Dilation with center at origin: (x', y')=k(x, y). So if S is (0, 6) and S' is (0, 2), then 2 = k*6 => k = 2/6 = 1/3. Alternatively, take the base. Original base TV: from x=-3 to x = 3, length 6. Dilated base T'V': from x=-1 to x = 1, length 2. So scale factor 2/6 = 1/3. So the scale factor is 1/3.

Step2: Confirm scale factor

Scale factor k is given by \( k=\frac{\text{Length of image}}{\text{Length of original}} \). For the y - coordinate (or x - coordinate for base), taking the y - coordinate of S (original) and S' (image): if S=(0,6) and S'=(0,2), then \( k = \frac{2}{6}=\frac{1}{3} \). For the base, original base length (TV) is 6 (from x=-3 to x = 3, distance is 6), image base length (T'V') is 2 (from x=-1 to x = 1, distance is 2), so \( k=\frac{2}{6}=\frac{1}{3} \).

Answer:

\(\frac{1}{3}\) (corresponding to the option \(\frac{1}{3}\))