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 ) .
The scale factor is [ ]
To find the scale factor, we can compare the distances between the corresponding vertices of the two triangles.
The distance between point A(2, 1) and the origin (0, 0) is √(2^2 + 1^2) = √5.
The distance between point A'(4, 2) and the origin (0, 0) is √(4^2 + 2^2) = √20 = 2√5.
The scale factor for the x-coordinate is 2√5 / √5 = 2.
Similarly, we can compare the distances for point B and point C:
- For point B, the distance ratio is -2√5 / -2√5 = 1.
- For point C, the distance ratio is -3√5 / -3√5 = 1.
Since the scale factor is the same for all the vertices, we can conclude that the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 2.