5. 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).
To find the scale factor of the dilation, we can compare the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
1. Find the distance between points A and B in triangle ABC.
Using the distance formula:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
AB = √[(-2 - 2)^2 + (2 - 1)^2]
AB = √[(4)^2 + (1)^2]
AB = √[16 + 1]
AB = √17
2. Find the distance between points A' and B' in triangle A'B'C'
Using the distance formula:
A'B' = √[(x2 - x1)^2 + (y2 - y1)^2]
A'B' = √[(-4 - 4)^2 + (4 - 2)^2]
A'B' = √[(-8)^2 + (2)^2]
A'B' = √[64 + 4]
A'B' = √68
A'B' = 2√17
3. The scale factor is the ratio of the corresponding side lengths:
scale factor = A'B' / AB
scale factor = (2√17) / √17
scale factor = 2
Therefore, the scale factor for the dilation of triangle ABC with the center of dilation at the origin (0,0) is 2.