Question

Triangle STV was dilated with the origin as the center of dilation to form Triangle S'T'V'. On a coordinate plane, triangles S T V and S prime T prime V prime are shown. Triangle S T V has points (0, 6), (negative 3, negative 6), and (3, negative 6). Triangle S prime T prime V prime has points (0, 2), (negative 1, negative 2) and (1, negative 2). What is the scale factor of the dilation? One-third Two-thirds Three-halves Three

50 POINTS!!!!!

Answer 1

the scale factor of the dilation is 17/51, which is equivalent to 1/3. Therefore, the scale factor is one-third.

Explanation:

To find the scale factor of the dilation, you can compare the corresponding side lengths of the original triangle STV and the dilated triangle S' T' V'. The scale factor (k) is given by the ratio of the lengths of corresponding sides.

Let's compare the lengths of the sides:

1. Side ST (original) and S'T' (dilated):

ST = sqrt((3 - 0)^2 + (-6 - 6)^2) = sqrt(3^2 + 12^2) = sqrt(9 + 144) = sqrt(153)

S'T' = sqrt((1 - 0)^2 + (-2 - 2)^2) = sqrt(1^2 + 4^2) = sqrt(1 + 16) = sqrt(17)

2. Side SV (original) and S'V' (dilated):

SV = sqrt((3 - 0)^2 + (-6 - (-6))^2) = sqrt(3^2 + 0^2) = sqrt(9) = 3

S'V' = sqrt((1 - 0)^2 + (-2 - (-2))^2) = sqrt(1^2 + 0^2) = sqrt(1) = 1

Now, let's find the ratio of the lengths of corresponding sides:

Scale Factor (k) = (Length of S'T') / (Length of ST) = sqrt(17) / sqrt(153)

To simplify this fraction, multiply both the numerator and denominator by sqrt(153):

k = (sqrt(17) / sqrt(153)) * (sqrt(153) / sqrt(153))

k = sqrt(17 * 153) / 153

Now, let's calculate the value of k:

k = sqrt(2601) / 153

k = 51 / 153

Now, simplify the fraction:

k = (51 / 3) / (153 / 3)

k = 17 / 51

So, the scale factor of the dilation is 17/51, which is equivalent to 1/3. Therefore, the scale factor is one-third.