Question

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

Answer 1

To determine the scale factor for the dilation from triangle ABC to triangle A′B′C′ with respect to the origin, we need to find the ratio of the corresponding coordinates of the corresponding vertices.

We can do this by comparing the coordinates of corresponding vertices in triangles ABC and A′B′C′.

1. Vertex A to A′:

   • A: (2, 1)
   • A′: (4, 2)

   To find the scale factor \( k \): \[ k = \frac{A'{x}}{A{x}} = \frac{4}{2} = 2 \] and \[ k = \frac{A'{y}}{A{y}} = \frac{2}{1} = 2 \]

2. Vertex B to B′:

   • B: (−2, 2)
   • B′: (−4, 4)

   Again, we calculate the scale factor: \[ k = \frac{B'{x}}{B{x}} = \frac{-4}{-2} = 2 \] and \[ k = \frac{B'{y}}{B{y}} = \frac{4}{2} = 2 \]

3. Vertex C to C′:

   • C: (−3, −1)
   • C′: (−6, −2)

   Finally, we check the scale factor: \[ k = \frac{C'{x}}{C{x}} = \frac{-6}{-3} = 2 \] and \[ k = \frac{C'{y}}{C{y}} = \frac{-2}{-1} = 2 \]

In all cases, the scale factor \( k \) is consistent. Thus, the scale factor for the dilation of triangle ABC to triangle A′B′C′ is:

\[ \boxed{2} \]