Question

Triangle STV was dilated with the origin as the center of dilation to form Triangle S'T'V'.

On a coordinate plane, triangles S T V and S prime T prime V prime are shown. Triangle S T V has points (0, 6), (negative 3, negative 6), and (3, negative 6). Triangle S prime T prime V prime has points (0, 2), (negative 1, negative 2) and (1, negative 2).

What is the scale factor of the dilation?

One-third
Two-thirds
Three-halves
Three

Answer 1

To find the scale factor of the dilation from Triangle STV to Triangle S'T'V', we can compare the coordinates of corresponding points from the original triangle and the dilated triangle.

Let's examine the points:

1. From Triangle STV:
- Point S: (0, 6)
- Point T: (-3, -6)
- Point V: (3, -6)

2. From Triangle S'T'V':
- Point S': (0, 2)
- Point T': (-1, -2)
- Point V': (1, -2)

To find the scale factor for each corresponding pair of points, we can calculate the ratio of the coordinates of a point in Triangle S'T'V' to the corresponding point in Triangle STV.

Starting with point S and S':
- For S (0, 6) and S' (0, 2):
- The ratio for y-coordinates: \( \frac{2}{6} = \frac{1}{3} \)

Next, for point T and T':
- For T (-3, -6) and T' (-1, -2):
- The ratio for x-coordinates: \( \frac{-1}{-3} = \frac{1}{3} \)
- The ratio for y-coordinates: \( \frac{-2}{-6} = \frac{1}{3} \)

Finally, for point V and V':
- For V (3, -6) and V' (1, -2):
- The ratio for x-coordinates: \( \frac{1}{3} \)
- The ratio for y-coordinates: \( \frac{-2}{-6} = \frac{1}{3} \)

In each case, the scale factor is consistently \(\frac{1}{3}\).

Thus, the scale factor of the dilation is **one-third**.