For the line segment AB, one endpoint is A (2,5) and the midpoint is M (-2,0).?

Find the coordinates of the Endpoint B.


I would like to know the steps to solving this question. Examples from the web are appreciated.

For the line segment AB, one endpoint is A (2,5) and the midpoint is M (-2,0).?
You want to find what y is when x = -6





Find the equation of the line going through the two points.
Reply:If you know the midpoint fomula you can work backwards from that.





((x1 + x2) / 2 , (y1 + y2) / 2) = midpoint





so you get (2 + x2)/2 = -2 to find the x coordinate of B





if you solve fofr x2 you get x2 = -6





then for y2, (5 + y2) / 2 = 0, so y2 = -5





and the coordinates for B are (-6,-5)
Reply:Point B would be (-4,-5) U would subtract 2 from the x value and 5 from the y value b/c the midpoint is 2 and 5 units away from point A. It would make sense 2 have point M be in the middle.





I'm sorry if this doesn't make any sense. But I hope it helps u.
Reply:Let the point B be (c,d). Then using the given that M( - 2,0) is the mid point of A(2,5) and B(c,d), we get:


(c+2)/2 = - 2 which gives c= -6.


(d+5)/2 = 0 which gives d = - 5.


So the point B is ( -6, -5 ).


Verifying OK


Now length of AB = sqrt (64+100) = sqrt(164) = 2 sqrt (41).


Or find the length of AM and double it.
Reply:Point A has (x,y) coordinates (2,5).


Point M has (x,y) coordinates (-2,0).





One way to solve this is to get the equation of the line that passes through this line segment. Once we know the distance AM, we can find the coordinates of B since AM = BM.





The equation of a line is given by : y = mx + b


( m is slope, b is y -intercept )








Let's find the slope first: This can be done using two points: A and M





The way to find the slope between A (x1,y1) and M(x2,y2) is by:





m = (x1-x2)/(y1-y2) = (2 - (-2) ) / ( 5 - 0 ) = 4 / 5





Now lets find the y-intercept: Use either point A or M for this. M is easier since its y component is 0.





y = (4/5)x + b


0 = (4/5)(-2) + b


8/5 = b





y = (4/5)x + 8/5 is the equation of the line segment passing thru AB.





Now we need to know the distance AM. This can be found by utilizing their coordinates. x changes from 2 and goes to -2. (difference of -4) y changes from 5 and goes to 0 (difference of 5)





Thus B = Mx - 4 , My - 5 = (-6, -5)