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 answer

To find the scale factor for the dilation of triangle ABC to triangle A′B′C′ with a center of dilation at the origin (0,0), we can use the relationship between the original coordinates and the coordinates after dilation.

For triangle ABC:

  • Point A (−2, 4)
  • Point B (−6, 8)
  • Point C (−2, 4)

For triangle A′B′C′:

  • Point A′ (−1, 2)
  • Point B′ (−3, 4)
  • Point C′ (−1, 2)

We calculate the scale factor \( k \) for each vertex by using the formula:

\[ k = \frac{\text{New Coordinate}}{\text{Old Coordinate}} \]

Let’s calculate the scale factor for point A and point A′:

For point A:

  • A = (−2, 4)
  • A′ = (−1, 2)

Calculating the x-coordinate scale factor: \[ k_x = \frac{-1}{-2} = \frac{1}{2} \]

Calculating the y-coordinate scale factor: \[ k_y = \frac{2}{4} = \frac{1}{2} \]

Both x and y coordinates give the same scale factor of \( \frac{1}{2} \).

Now let's check point B and point B′ to verify that the same scale factor applies:

For point B:

  • B = (−6, 8)
  • B′ = (−3, 4)

Calculating the x-coordinate scale factor: \[ k_x = \frac{-3}{-6} = \frac{1}{2} \]

Calculating the y-coordinate scale factor: \[ k_y = \frac{4}{8} = \frac{1}{2} \]

So, for point B the scale factor is also \( \frac{1}{2} \).

Since the dilations for points A, B, and C yield the same scale factor, we can conclude that the overall scale factor for the dilation of triangle ABC to triangle A′B′C′ is:

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