1. a) The scale factor that transform ΔKLP to ΔK'L'P' is a dilation of 3/2
b) The rule represented is that if the scale factor is larger than 1, the image formed is larger than the preimage
2. a) The scale factor of the dilation is 3/4
b) The rule represented by the dilation is a scale factor of less than 1 produces an image with a distance from a given point which is less than the distance of the preimage from the given point
Step-by-step explanation:
1. The coordinates of the triangle ΔKLP are;
K(-4, 2), L(2, 6), P(4, -6)
The coordinates of the points of triangle ΔK'L'P' are;
K'(-6, 3), L'(3, 9), P'(6, -9)
The length of segment KL = √((2 - (-4))² + (6 - 2)²) = 2·√13
The length of segment LP = √((4 - 2)² + (-6 - 6)²) = 2·√37
The length of segment KP = √((4 - (-4))² + (-6 - 2)²) = 8·√2
The length of segment K'L' = √((3 - (-6))² + (9 - 3)²) = 3·√13
The length of segment L'P' = √((6 - 3)² + (-9 - 9)²) = 3·√37
The length of segment K'P' = √((6 - (-6))² + (-9 - 3)²) = 12·√2
Therefore, K'L'/KL = 3·√13/(2·√13) = L'P'/LP = (3·√37/(2·√37)) = K'P'/KP =(12·√2/(8·√2)) = 3/2
Therefore, the scale factor that transform ΔKLP to ΔK'L'P' is a dilation of 3/2
The rule represented is that if the scale factor is larger than 1, the image formed (here ΔK'L'P') is larger than the preimage (ΔKLP)
2. The location of the preimage, W(-12, -4), the location of the image after the dilation W'(-9, -3)
Taking the origin O(0, 0), as the center of dilation, we have;
The length of the preimage W(-12, -4) from the origin = √((-12 - 0)² + (-4 - 0)²) = 4·√10
The length of the image W'(-9, -3) from the origin = √((-9 - 0)² + (-3 - 0)²) = 3·√10
Therefore, the scale factor of the dilation = W'O/WO = 3·√10/(4·√10) = 3/4
The scale factor of the dilation = 3/4
The rule represented by the dilation is that for a scale factor less than 1, the distance between the point of the image from a given point is less than the distance of the point of the preimage from the same point
