Dilations Quick Check

Question

5 of 55 of 5 Items
Question
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

2
2

−12
negative Start Fraction 1 over 2 End Fraction

12
Start Fraction 1 over 2 End Fraction

−2

Answer 1

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 formula for dilation:

\[ A' = k \cdot A \] where \(k\) is the scale factor, \(A\) is the original point, and \(A'\) is the dilated point.

Let's find the scale factor using one of the points, point A. The original point A is \((-2, 4)\) and the dilated point A' is \((-1, 2)\).

Using the x-coordinates: \[ -1 = k \cdot (-2) \] Solving for \(k\): \[ k = \frac{-1}{-2} = \frac{1}{2} \]

Now using the y-coordinates: \[ 2 = k \cdot 4 \] Solving for \(k\): \[ k = \frac{2}{4} = \frac{1}{2} \]

Both calculations confirm that the scale factor is \( \frac{1}{2} \).

Thus, the answer to the question is:

\[ \frac{1}{2} \] (which corresponds to the response option: \( \frac{1}{2} \))