Triangle ABC has vertices point A (−2,4) , point B (−6,8) , and point C (−2,4) . After dilation, triangle A′B′C′ has vertices point A′ (−1,2) , point B′ (−3,4) , and point C′ (−1,2) . Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) . (1 point) Responses −12 negative Start Fraction 1 over 2 End Fraction 12 Start Fraction 1 over 2 End Fraction −2 negative 2 2

Question

GPT-4o mini · Answer

To determine the scale factor of the dilation, we can use the formula for calculating the scale factor \( k \) of a dilation centered at the origin.

The formula for the scale factor \( k \) is: \[ k = \frac{x'}{x} \] where \( (x, y) \) are the coordinates of the original point and \( (x', y') \) are the coordinates of the dilated point.

We will calculate \( k \) using one pair of corresponding points:

Using point A:

Original point A: \( A(-2, 4) \)
Dilated point A': \( A'(-1, 2) \)

For the x-coordinates: \[ k = \frac{-1}{-2} = \frac{1}{2} \]

For the y-coordinates: \[ k = \frac{2}{4} = \frac{1}{2} \]

Both calculations yield \( k = \frac{1}{2} \).

We can check this with another pair of points to confirm it: Using point B:

Original point B: \( B(-6, 8) \)
Dilated point B': \( B'(-3, 4) \)

For the x-coordinates: \[ k = \frac{-3}{-6} = \frac{1}{2} \]

For the y-coordinates: \[ k = \frac{4}{8} = \frac{1}{2} \]

Again, we get \( k = \frac{1}{2} \).

Since the scale factor is consistent for both points and is the same across both sets of points, we conclude that the scale factor for the dilation of triangle ABC to triangle A'B'C' is:

\[ \boxed{\frac{1}{2}} \]