Geometry style question?

Question

Hi, This question is bugging me, I cant do the last part.

A is the point (2,3) and B is the point (14,8)

P and Q are points such that APBQ forms a square, Find the co-ordinates of P and Q

well I started to do this by finding the length of AB as 13 as AB = PQ. I also found the midpoint of AB to be (8, 5.5) and the equation of AB is also 12y= 5x+26

Now I know the diagonals of a square meet at right angles and at the midpoint of both bisectors, but I cannot figure how to get the co-ordinates of the other two pairs,

Any help is much appreciated

Answer 1

You're thinking too hard! :) Use the slope of the line, and of its perpendicular.

B is 12 units right and 5 up from A. You're right that the midpoint is the center of the square, which is (8, 5.5). B, from the midpoint, is 6 units right and 2.5 up. For one of the other corners of your square, go 2.5 left and 6 up... for the other, go 2.5 right and 6 down.

P and Q (either order), are (5.5, 11.5) and (10.5, -0.5).

Answer 2

If two lines are perpendicular to each other, and the slope of the first one is m, the the slope of the other one is -1/m. AB has a slope of 5/12, so you need to find a line through (8 , 5.5) with slope -12/5 = -2.4.

I think you can probably take it from there. Let me know if you're still having trouble.

Answer 3

Write the equation of the diagonal
y-8= 5/7(x-14) so its slope is 5/12
Through A draw a line which forms 45 degrees with the diagonal
(m´-m)/(1+mm´)= 1 so you get m´which is the slope of one side.
(if you put m-m´you´ll get the slope of the other side)
Intercept both sides with the bisector of AB
which passes through (8,11/5) and has slope -12/5
So you´ll get P and Q