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 ).(1 point)

Question

The scale factor is
.

Answer 1

To find the scale factor, we can compare the distances between the corresponding points of the two triangles.

The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the distances between the corresponding points:

Distance between A and A':
d(A, A') = sqrt((4 - 2)^2 + (2 - 1)^2) = sqrt(4 + 1) = sqrt(5)

Distance between B and B':
d(B, B') = sqrt((-4 - (-2))^2 + (4 - 2)^2) = sqrt(4 + 4) = sqrt(8) = 2*sqrt(2)

Distance between C and C':
d(C, C') = sqrt((-6 - (-3))^2 + (-2 - (-1))^2) = sqrt(9 + 1) = sqrt(10)

To find the scale factor, we divide the distances between the corresponding points of the dilated triangle by the distances between the corresponding points of the original triangle.

Scale factor = d(A, A') / d(A, A') = sqrt(5) / 1 = sqrt(5)

Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is sqrt(5).

Answer 2

nope try again

Answer 3

Apologies for the mistake. Let's try again.

To find the scale factor, we can compare the lengths of the corresponding sides of the two triangles. Since the center of dilation is at the origin (0, 0), we can calculate the scale factor by dividing the length of a side of the dilated triangle by the length of the corresponding side of the original triangle.

Let's calculate the length of the corresponding sides:

Side AB:
Length of AB = sqrt((2 - (-2))^2 + (1 - 2)^2) = sqrt(4 + 1) = sqrt(5)

Length of A'B' = sqrt((4 - (-4))^2 + (2 - 4)^2) = sqrt(64 + 4) = sqrt(68) = 2sqrt(17)

Scale factor for side AB = Length of A'B' / Length of AB = 2sqrt(17) / sqrt(5) = 2sqrt(17) * sqrt(5) / 5 = 2sqrt(85) / 5 = (2/5)sqrt(85)

Similarly, we can calculate the scale factors for the other sides:

Scale factor for side BC = (Length of B'C') / (Length of BC) = (2sqrt(17)) / (sqrt(5)) = 2sqrt(17) * sqrt(5) / 5 = 2sqrt(85) / 5 = (2/5)sqrt(85)

Scale factor for side AC = (Length of A'C') / (Length of AC) = (4sqrt(17)) / (sqrt(5)) = 4sqrt(17) * sqrt(5) / 5 = 4sqrt(85) / 5 = (4/5)sqrt(85)

Since the scale factor should be the same for all sides in a dilation, we'll take the scale factor as the average of the individual scale factors.

Average scale factor = ((2/5)sqrt(85) + (2/5)sqrt(85) + (4/5)sqrt(85)) / 3 = (8/5)sqrt(85) / 3 = (8/15)sqrt(85)

Therefore, the scale factor for the dilation of triangle ABC with a center point of dilation at the origin is (8/15)sqrt(85).