coordinate geometry question!!!

[ashoni]

hey guys, im having difficulties trying to find a method to solve a question from my methods textbook.. was hoping one of youse could help . Heres the question..

Given two points A(-1,1) and B(7,7) and another point P(x,y) on the line, find the coordinates of P so that PA=PB=5.

please show working out if possible, thanks

[Phy124]

The coordinate should be (3,4) I'm pretty sure.

This is probably the only way I can show lots of working out/proper explanation (I hope I didn't over complicate things for you):

http://i43.tinypic.com/2naobp0.jpg

[ashoni]

yeep thats the answer thanks for the help Phy124!

[Phy124]

No worries, always glad to help

[ashoni]

hey, sorry to bother you again, but i dont understand the wording of this question.

A line makes an angle of 45 degrees with the x-axis and passes through the midpoint of the line joining the points A(3,1) and B(6,2).
 (a) Find the gradient of the line.
 (b) Find the equation of the line.
 (c) Another line that has a y-intercept of 3 is perpendicular to the above line. Find the point of intersection.

I have solved part (a) through the tan of the angle, tan(45) which is obviously 1, but i dont know how to do part (b), dont worry about part (c), i think ill be able to work it out after i get how to do part (b)

[brightsky]

okay, midpoint of A and B is (4.5, 1.5).
we already know that the line has gradient 1.
hence by the 'gradient point' formula or whatever it's called:
y - 1.5 = x - 4.5
y = x - 3

[dc302]

To find the equation of the line, simply use y-y1 = m(x-x1), where m=(y1-y2)/(x1-x2). m is the gradient of course, and you should be able to find points for x1,y1.

[ashoni]

okay, midpoint of A and B is (4.5, 1.5).
we already know that the line has gradient 1.
hence by the 'gradient point' formula or whatever it's called:
y - 1.5 = x - 4.5
y = x - 3

yeah, i tried that way but the answer is 3y-x-5=0 D:

[ashoni]

wait, dont worry thats the right answer
was looking at the wrong question the whole time -.- LOL

[Special At Specialist]

1. If a point is on a line with 2 other points and the distance from it to either one of the other points is the same, then that point MUST be halfway between the other two.
In other words, given the points A(-1,1) and B(7,7):
Halfway between -1 and 7 is 3 --- x-coordinate.
Halfway between 1 and 7 is 4 --- y-coordinate.
(3, 4)

2. a) m = tan(θ) = dy/dx = rise/run = (y2 - y1) / (x2 - x1)
Using m = tan(θ)
m = tan(45 deg)
m = 1

b) First find the midpoint of (x1, y1) and (x2, y2) which is ((x1 + x2)/2, (y1 + y2)/2) where (x1, y1) = (3, 1) and (x2, y2) = (6, 2)
Midpoint = ((3 + 6)/2, (1 + 2)/2)
Midpoint = (9/2, 3/2)
Using the formula (y - y1) = m(x - x1) where m = 1 and (x1, y1) = (9/2, 3/2)
y - 3/2 = 1(x - 9/2)
y = x - 9/2 + 3/2
y = x - 3

c) y-intercept of 3 means it has the point (0, 3):
Perpendicular has gradient -1/m = -1/1 = -1
Now use the formula (y - y1) = m(x - x1) where m = -1 and (x1, y1) = (0, 3)
y - 3 = -1(x - 0)
y = -x + 3