Question

Triangle ABC with vertices at A(−1, −1), B(1, 1), C(0, 1) is dilated to create triangle A′B′C′ with vertices at A′(−3, −3), B′(3, 3), C′(0, 3). Determine the scale factor used.

1
one half
3
1/3

Answer 1

To determine the scale factor used in the dilation from triangle ABC to triangle A′B′C′, we can use the coordinates of the vertices.

1. First, calculate the distance from the origin (0, 0) to each vertex of triangle ABC.

- For point A(-1, -1):
\[
d_A = \sqrt{(-1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

- For point B(1, 1):
\[
d_B = \sqrt{(1-0)^2 + (1-0)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

- For point C(0, 1):
\[
d_C = \sqrt{(0-0)^2 + (1-0)^2} = \sqrt{0 + 1} = 1
\]

2. Next, calculate the distances from the origin to each vertex of triangle A′B′C′.

- For point A′(−3, −3):
\[
d_{A'} = \sqrt{(-3-0)^2 + (-3-0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

- For point B′(3, 3):
\[
d_{B'} = \sqrt{(3-0)^2 + (3-0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

- For point C′(0, 3):
\[
d_{C'} = \sqrt{(0-0)^2 + (3-0)^2} = \sqrt{0 + 9} = 3
\]

3. Determine the scale factor \( k \) for the dilation to find the relationship between the distances. The scale factor can be calculated using the distances we found.

Using point A and A′:
\[
k = \frac{d_{A'}}{d_A} = \frac{3\sqrt{2}}{\sqrt{2}} = 3
\]

Using point C and C′:
\[
k = \frac{d_{C'}}{d_C} = \frac{3}{1} = 3
\]

From both vertex calculations, we find that the scale factor is consistent and equal to 3.

Thus, the scale factor used is \( \boxed{3} \).