9514 1404 393
Answer:
y = 21
Explanation:
In straightforward fashion, we can find the midpoints of the two segments, and set the distance between them equal to 10.
M = midpoint of AB
M = (A+B)/2 = ((-1, 0) +(-3, 6))/2 = (-4, 6)/2 = (-2, 3)
N = midpoint of CD
N = (C+D)/2 = ((3, 1) +(5, y))/2 = (8, 1+y)/2 = (4, (1+y)/2)
Then the distance MN is given by the distance formula.
|N-M| = 10 = √((4 -(-2))² +((1+y)/2 -3)²) . . . . . distance formula*
100 = 36 +((1+y)/2 -3)² . . . . . square both sides
64 = ((1 +y)/2 -3)² . . . . . . . . . subtract 36
8 = (1 +y)/2 -3 . . . . . . . . . . . . (positive) square root
11 = 1+y/2 . . . . . . . . add 3
22 = 1+y . . . . . . . . . multiply by 2
21 = y . . . . . . . . . . . subtract 1
_____
* The distance formula is an application of the Pythagorean theorem. Basically, the differences of x- and y-coordinates are taken to be the legs of a right triangle. They length of the hypotenuse is the distance between the two points.
d² = (x2 -x1)² +(y2 -y1)² . . . . . . . from the Pythagorean theorem
d = √((x2 -x1)² +(y2 -y1)²) . . . . . the usual distance formula