Given a scale factor of 2, find the coordinates for the dilation of the line segment with endpoints (–1, 2) and (3, –3)
let the given points be A(-1,2) and B(3,-3)
So the midpoint of AB is M(1,-1/2)
Extend AB to P so that MP = AB
and extend BA to Q so that MQ = AB
it is easy to see that PQ = 2AB
to find P, consider B to be the midpoint of BP
for the x and y of P:
(x+1)/2 = 3 and (y-1/2)/2 = -3
x+1 = 6 and y-1/2 = -6
x = 5 and y = -11/2
P is (5,-11/2) or (5, -5.5)
in the same way ...
Q is (-3 , 9/2) or (-3, 4.5)
check:
AB = √(4^2 + (-5)^2) = √41
PQ = √( 8^2 + 10^2) = √164 = 2√41
so PQ = 2AB and also A,B,P, and Q are collinear
To find the coordinates for the dilation of a line segment, we need to perform the following steps:
1. Find the midpoint of the line segment.
2. Calculate the distance between each endpoint and the midpoint.
3. Multiply the distances by the scale factor.
4. Add or subtract the scaled distances from the midpoint coordinates to find the new coordinates.
Let's follow these steps to find the coordinates for the dilation of the line segment with endpoints (–1, 2) and (3, –3) using a scale factor of 2.
Step 1: Find the midpoint
To find the midpoint, we average the x-coordinates and the y-coordinates of the endpoints separately.
Midpoint coordinates (x, y) = ((–1 + 3) / 2, (2 + (–3)) / 2)
= (2 / 2, –1 / 2)
= (1, –1/2)
So, the midpoint of the line segment is (1, –1/2).
Step 2: Calculate the distance to each endpoint from the midpoint
We can use the distance formula to find the distance between each endpoint and the midpoint.
Distance between two points (x1, y1) and (x2, y2) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance from endpoint (–1, 2) to the midpoint (1, –1/2):
sqrt((1 - (–1))^2 + ((–1/2) - 2)^2)
= sqrt((2)^2 + (–5/2)^2)
= sqrt(4 + 25/4)
= sqrt(16/4 + 25/4)
= sqrt(41/4)
Distance from endpoint (3, –3) to the midpoint (1, –1/2):
sqrt((3 - 1)^2 + ((–3) - (–1/2))^2)
= sqrt((2)^2 + (–5/2 - (–3))^2)
= sqrt((2)^2 + (–5/2 + 6))^2)
= sqrt(4 + 9/4)
= sqrt(16/4 + 9/4)
= sqrt(25/4)
Step 3: Multiply the distances by the scale factor
Scale factor = 2
Scaled distance from endpoint (–1, 2) to the midpoint: sqrt(41/4) * 2 = 2 * sqrt(41)/2 = sqrt(41)
Scaled distance from endpoint (3, –3) to the midpoint: sqrt(25/4) * 2 = 2 * sqrt(25)/2 = 5
Step 4: Add or subtract the scaled distances from the midpoint coordinates to find the new coordinates
Dilated endpoint coordinates:
For (–1, 2): (1 – sqrt(41), –1/2) and (1 + sqrt(41), –1/2)
For (3, –3): (1 – 5, –1/2) and (1 + 5, –1/2)
So, the coordinates for the dilation of the line segment with endpoints (–1, 2) and (3, –3) using a scale factor of 2 are:
(1 – sqrt(41), –1/2), (1 + sqrt(41), –1/2), (–4, –1/2), and (6, –1/2).