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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1 answer

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.