Question

Triangle ABC has vertices A (-3, 1), B (0, 3), and C (3, 0) and is dilated by a factor of 1/2 using point (-2, 1) as the point of dilation. The dilated triangle names A’ B’ C’. What are the coordinates of the vertices of the resulting triangle?

Answer 1

After performing a dilation with a scale factor of 1/2 and using point (-2, 1) as the center, the coordinates of the dilated triangle A'B'C' are A'(-2.5, 1), B'(-1, 2), and C'(0.5, 0.5).

Step-by-step explanation:

The process of finding the resulting vertices after a dilation involves using the coordinates of the original vertices and the center of dilation. For each vertex, we find the difference between the vertex and the center of dilation, multiply by the scale factor, and add the results back to the center of dilation's coordinates.

With scale factor ½ and center of dilation at (-2, 1), the coordinates of vertices A' B' C' are calculated as follows:

• For A' from A (-3, 1), the differences are dx = -3 - (-2) = -1, dy = 1 - 1 = 0. Multiplying by ½: (½)(-1), (½)(0) = (-½, 0). Adding these to the center coordinates: A' = (-2 - ½, 1 + 0) = (-2.5, 1).
• For B' from B (0, 3), the differences are dx = 0 - (-2) = 2, dy = 3 - 1 = 2. Multiplying by ½: (½)(2), (½)(2) = (1, 1). Adding these to the center coordinates: B' = (-2 + 1, 1 + 1) = (-1, 2).
• For C' from C (3, 0), the differences are dx = 3 - (-2) = 5, dy = 0 - 1 = -1. Multiplying by ½: (½)(5), (½)(-1) = (2.5, -0.5). Adding these to the center coordinates: C' = (-2 + 2.5, 1 - 0.5) = (0.5, 0.5).

Therefore, the coordinates of the vertices of the dilated triangle A' B' C' are A'(-2.5, 1), B'(-1, 2), and C'(0.5, 0.5).

Answer 2

Dilations change a shape's size by a number (better known as a factor). You multiply the factor by the numbers for each point:

• A dilated from (-2,1) = [A' = (-3.5,1.5)]
• B dilated from (-2,1) = [B' = (-2,2.5)]
• C dilated from (-2,1) = [C' = (-0.5,1)]