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
the scale factor is [ ]
To find the scale factor, we can compare the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
Let's start with side AB. The original length of AB is the distance between points A and B:
d(AB) = √((x2-x1)^2 + (y2-y1)^2) = √((-2-2)^2 + (2-1)^2) = √((-4)^2 + (1)^2) = √(16 + 1) = √17.
The length of A'B' is the distance between points A' and B':
d(A'B') = √((x2-x1)^2 + (y2-y1)^2) = √((-4-4)^2 + (4-2)^2) = √((-8)^2 + (2)^2) = √(64 + 4) = √68 = 2√17.
So the scale factor for side AB is 2.
Similarly, we can find the scale factors for the other sides:
Scale factor for side BC:
d(BC) = √((-3+2)^2 + (-1-2)^2) = √((-1)^2 + (-3)^2) = √(1 + 9) = √10.
d(B'C') = √((-6+4)^2 + (-2-4)^2) = √((-2)^2 + (-6)^2) = √(4 + 36) = √40 = 2√10.
Scale factor for side CA:
d(CA) = √((2+3)^2 + (1+1)^2) = √((5)^2 + (2)^2) = √(25 + 4) = √29.
d(C'A') = √((4+6)^2 + (2+2)^2) = √((10)^2 + (4)^2) = √(100 + 16) = √116 = 2√29.
Since all three pairs of corresponding side lengths have the same scale factor of 2, we can conclude that the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is 2.