Triangle ABC with vertices at A(−3, −3), B(3, 3), C(0, 3) is dilated to create triangle A′B′C′ with vertices at A′(−12, −12), B′(12, 12), C′(0, 12). Determine the scale factor used.

9
one ninth
4
one fourth

1 answer

To determine the scale factor used for the dilation from triangle ABC to triangle A'B'C', we can look at the coordinates of corresponding points before and after dilation.

For point A to A':

  • Point A: A(-3, -3)
  • Point A': A'(-12, -12)

To find the scale factor, we divide the coordinates of A' by the coordinates of A. Let's take one coordinate (either x or y, they will be the same for this calculation):

\[ \text{Scale Factor} = \frac{A'}{A} = \frac{-12}{-3} = 4 \]

We can verify this by checking other points:

For point B to B':

  • Point B: B(3, 3)
  • Point B': B'(12, 12)

\[ \text{Scale Factor} = \frac{B'}{B} = \frac{12}{3} = 4 \]

And for point C to C':

  • Point C: C(0, 3)
  • Point C': C'(0, 12)

\[ \text{Scale Factor} = \frac{C'}{C} = \frac{12}{3} = 4 \quad (\text{0 remains 0, so only y-coordinate matters here}) \]

Thus, the scale factor is consistently calculated to be 4.

The scale factor used for the dilation from triangle ABC to triangle A'B'C' is 4.