Question

Triangle ABC is dilated about the origin to create triangle A′B′C′.

Triangle ABC with vertices at A negative 6 comma negative 4, B negative 2 comma negative 4, and C negative 2 comma 2 and triangle A prime B prime C prime with vertices at A prime negative 9 comma negative 6, B prime negative 3 comma negative 6, and C prime negative 3 comma 3.

Determine the scale factor used to create the image.

two fifths
1.5
one half
2.5

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 one of the vertices.

Let's take vertex A and A' as examples. The coordinates of A and A' are:

- \( A(-6, -4) \)
- \( A'(-9, -6) \)

The scale factor \( k \) can be calculated by comparing the distances from the origin to the respective points.

To find the distances:

1. **Distance from the origin to A:**
\[
d_A = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
\]

2. **Distance from the origin to A':**
\[
d_{A'} = \sqrt{(-9)^2 + (-6)^2} = \sqrt{81 + 36} = \sqrt{117} = 3\sqrt{13}
\]

Next, we can find the scale factor \( k \) by dividing the distance to A' by the distance to A:

\[
k = \frac{d_{A'}}{d_A} = \frac{3\sqrt{13}}{2\sqrt{13}} = \frac{3}{2} = 1.5
\]

Thus, the scale factor used to create triangle A'B'C' from triangle ABC is:

\[
\boxed{1.5}
\]