Question

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 )

Answer 1

To find the scale factor, we can compare the distances between corresponding points of the original triangle ABC and the dilated triangle A'B'C'.

The distance between points A and B in triangle ABC is given by the distance formula:

sqrt((x2 - x1)^2 + (y2 - y1)^2)

= sqrt(((-2) - 2)^2 + (2 - 1)^2)

= sqrt((-4)^2 + 1^2)

= sqrt(16 + 1)

= sqrt(17)

The distance between points A' and B' in triangle A'B'C' is:

sqrt(((-4) - 4)^2 + (4 - 2)^2)

= sqrt((-8)^2 + 2^2)

= sqrt(64 + 4)

= sqrt(68)

Therefore, the ratio of the distances between the corresponding points A'B' and AB is:

sqrt(68) / sqrt(17)

To simplify this ratio, we rationalize the denominator by multiplying the numerator and denominator by sqrt(17):

(sqrt(68) / sqrt(17)) * (sqrt(17) / sqrt(17))

= sqrt(68 * 17) / 17

= sqrt(1156) / 17

= 34 / 17

= 2

Therefore, the scale factor for the dilation of triangle ABC with a center point at the origin is 2.