Finding midpoints and endpoints?

Question

1. what would the coordinates of the midpoint be if the endpoints are (-7,2) and (3,0). 2. What would the coordinates of the other endpoint be if the midpoint is (3,0) and one endpoint is (4,1)?

Answer 1

Both of these problems are solved using the midpoint formula: ((x1+x2)/2, (y1 + y2)/2). You're basically averaging the two x-coordinates and the two y-coordinates to find the point in the middle, the midpoint.

1. Call (-7, 2) your first point (x1, y1) and (3, 0) your second point(x2, y2). Plug in to the midpoint formula to get:

((-7 + 3)/2 , (2 + 0)/2)
(-4/2 , 2/2)
(-2, 1)

2. This one we'll solve on coordinate at a time, since we're given the midpoint but not the second point (x2, y2)

X-coordinate:

3 = (4 + x2)/2
6 = (4 + x2) multiply both sides by 2 to eliminate fraction
2 = x2 subtract 4 from both sides.

Y-coordinate

0 = (1 + y2)/2
0 = (1 + y2) mult. by 2
-1 = y2 subt. 1

So your other endpoint is (2, -1). You can check your answer by plugging back into the midpoint formula.

Answer 2

midpoint rule is ( (x1 + x2)/2 ) , ( (y1 + y2)/2 )

So, the x coordinate midoint would be (-7 + 3) = -4, divide by 2 = -2.

y coordinate midpoint would (2 + 0) = 2, divided by 2, = 1.

midpoint coordinates are therefore (-2,1)

Your second question. Well thinking in terms of x, whats the distance between the endpoint x coordinate and the midpoint x coordinate? 3 and 4 right? Difference is one.

the midpoint is right smack in the middle of the two endpoints, so if the midpoint is at 3, an endpoint is at 4, then the other endpoint must be 2.

thinking in terms of y, the distance between endpoint and midpoint is 1. middle is 0, one end is one, the other end must be -1.

therefore the endpoint coordinates would be (2/-1)

Answer 3

mid point is 1/2 the x difference and 1/2 the y difference added to the one end point or subtracted from the other end point.

|∆ x| = |-7 - (+3) | /2 = 10/2 = 5 x units
|∆ y| = |2 - 0 | /2 = 2/2 = 1 y unit

Now, you add 5 to -7 or subtract 3 from 5 to get -2
this is your x, x=-2
for the y, you add 1 unit to 0 or subtract 1 from 2 either way you get 1, y=1
so your point is (-2,1)
it helps to sketch the first few problems on graph paper to see what is going on and you will get better at this after 3 or 4 problems.

The idea is to find out how much x varies from one extreme to the other and take half of that amount to get the x mid point and do the same for y.

For the second part do the reverse of the above, for x the mid point is given and one end point is unknown.

for x if one end point is 4 and the mid point is 3 ( then the change in x is one less, |∆ x|= 1) so another one less form the mid point of 3 gives you a 2 (3-1=2), the same for y, y went from 1 to 0 ( |∆ y|= 1) so one less than 0 is -1 (0-1= -1), this is the other end point's y coordinate.
your previously unknown end point is now (2,-1)

Answer 4

if (x1, y1) and (x2, y2) are end points, then mid points are given by
((x1 + x2)/2, (y1 + y2)/2)
1)
x1 = - 7, x2 = 3 : (x1 + x2)/2 = (-7 + 3)/2 = -4/2 = -2
y1 = 2, y2 = 0 ; (y1 + y2)/2 = (2 + 0)/2 = 2/2 = 1
mid point coordinates are (-2, 1)

2)
x1 = 4, y1 = 1 , mid point(3, 0)
3 = (4 + x2)/2; x2 + 4 = 6 ; x2 = 2
0 = (1 + y2)/2; y2 = -1
so other end coordinates are (2, - 1)

Answer 5

1. (-2,1)
2. (2,-1)