Triangle ABC has vertices point A (−2,4) , point B (−6,8) , and point C (−2,4) . After dilation, triangle A′B′C′ has vertices point A′ (−1,2) , point B′ (−3,4) , and point C′ (−1,2) . Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) . (1 point)
Responses
−12
negative Start Fraction 1 over 2 End Fraction
2
2
12
Start Fraction 1 over 2 End Fraction
−2
negative 2
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page 19 of 19
To find the scale factor of the dilation, we compare the distances between corresponding vertices of triangle ABC and triangle A'B'C'.
The distance between points A (-2,4) and A' (-1,2) is found using the distance formula:
sqrt((x2-x1)^2 + (y2-y1)^2)
= sqrt(((-1)-(-2))^2 + (2-4)^2)
= sqrt((1)^2 + (-2)^2)
= sqrt(1+4)
= sqrt(5)
The distance between points B (-6,8) and B' (-3,4) is:
sqrt(((-3)-(-6))^2 + (4-8)^2)
= sqrt((3)^2 + (-4)^2)
= sqrt(9 + 16)
= sqrt(25)
= 5
The distance between points C (-2,4) and C' (-1,2) is:
sqrt(((-1)-(-2))^2 + (2-4)^2)
= sqrt((1)^2 + (-2)^2)
= sqrt(1+4)
= sqrt(5)
Now, we compare these distances to find the scale factor:
sqrt(5) / sqrt(5) = 1
So, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) is 1.