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 for the dilation, we need to compare the corresponding side lengths of the original triangle and the dilated triangle.
Let's first find the side lengths of triangle ABC:
Side AB:
Using the distance formula, we get:
AB = √[(-2 - 2)^2 + (2 - 1)^2]
= √[(-4)^2 + 1]
= √[16 + 1]
= √17
Side BC:
Using the distance formula, we get:
BC = √[(-3 - (-2))^2 + (-1 - 2)^2]
= √[(-1)^2 + (-3)^2]
= √[1 + 9]
= √10
Side AC:
Using the distance formula, we get:
AC = √[(2 - (-3))^2 + (1 - (-1))^2]
= √[(5)^2 + (2)^2]
= √[25 + 4]
= √29
Now let's find the side lengths of triangle A'B'C':
Side A'B':
Using the distance formula, we get:
A'B' = √[(4 - (-4))^2 + (2 - 4)^2]
= √[(8)^2 + (-2)^2]
= √[64 + 4]
= √68
= 2√17
Side B'C':
Using the distance formula, we get:
B'C' = √[(−4 - (−6))^2 + (4 - (−2))^2]
= √[(-2)^2 + (6)^2]
= √[4 + 36]
= √40
= 2√10
Side A'C':
Using the distance formula, we get:
A'C' = √[(4 - (−6))^2 + (2 - (−2))^2]
= √[(10)^2 + (4)^2]
= √[100 + 16]
= √116
= 2√29
Now, comparing the corresponding side lengths of the two triangles, we can see that the scale factor is the ratio of the corresponding side lengths:
Scale Factor = (A'B' / AB) = (2√17 / √17) = 2
Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0, 0) is 2.