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 ) .(1 point)
To find the scale factor, we can compare the corresponding sides of the original triangle ABC and the dilated triangle A'B'C'.
The distance formula is used to find the length of a side in coordinate geometry. The distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the lengths of the sides in the original triangle ABC:
Side AB:
d(AB) = √((-2 - 2)^2 + (2 - 1)^2) = √(16 + 1) = √17
Side BC:
d(BC) = √((-3 - (-2))^2 + (-1 - 2)^2) = √(1 + 9) = √10
Side AC:
d(AC) = √((-3 - 2)^2 + (-1 - 1)^2) = √(25 + 4) = √29
Now we can find the lengths of the corresponding sides in the dilated triangle A'B'C':
Side A'B':
d(A'B') = √((-4 - 4)^2 + (4 - 2)^2) = √(64 + 4) = √68 = 2√17
Side B'C':
d(B'C') = √((-6 - (-4))^2 + (-2 - 4)^2) = √(4 + 36) = √40 = 2√10
Side A'C':
d(A'C') = √((-6 - 4)^2 + (-2 - 2)^2) = √(100 + 16) = √116 = 2√29
To find the scale factor, we can compare the corresponding side lengths:
Scale factor = (length in dilated triangle) / (length in original triangle)
For side A'B', the scale factor = (2√17) / (√17) = 2
For side B'C', the scale factor = (2√10) / (√10) = 2
For side A'C', the scale factor = (2√29) / (√29) = 2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 2.