Question

Dilations Quick Check
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

12
Start Fraction 1 over 2 End Fraction

−2
negative 2

2
2

−12
negative Start Fraction 1 over 2 End Fraction

Answer 1

To determine the scale factor for the dilation of triangle ABC to triangle A′B′C′, we can use the formula for dilation:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

where \((x, y)\) are the original coordinates, \((x', y')\) are the new coordinates, and \(k\) is the scale factor.

Taking point A as an example:
- Original point A: \((-2, 4)\)
- Dilated point A′: \((-1, 2)\)

Using the formula for dilation, we can set up the equations:
\[
-1 = k \cdot (-2) \quad \text{and} \quad 2 = k \cdot 4
\]

From the first equation:
\[
-1 = -2k \implies k = \frac{1}{2}
\]

From the second equation:
\[
2 = 4k \implies k = \frac{2}{4} = \frac{1}{2}
\]

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

Thus, the scale factor for the dilation of triangle ABC to triangle A′B′C′ is:

**Start Fraction 1 over 2 End Fraction**