Arghhh Coordinate geometry question

[ultraviolent]

Hi everyone. This is a question that was attempted in class but no on (including our teacher) could solve in the time we had left. It's not actually homework, but it's really getting on my nerves!

The quadrilateral ABCD, where A is (4,5) and C is (3, -2), is a square. Find the coordinates of B and D and the area of the square.

Many thanks,

- James

 

[Tide]

You can find the slope of line segment AC (a diagonal) as well as its length and its midpoint. The negative reciprocal of that slope is the slope of the other diagonal since you're working with a square. Write an equation of a straight line using the point-slope form using the midpoint and the slope of the second diagonal. Use that equation to find the two points on that line that are half the length of the diagonal from the midpoint. That gives you the points B & D.

Knowing the length of the diagonal you can easily find the area of a square.

 

[ultraviolent]

I already managed to do the first part, but how would you use the equation derived from this to calculate the points half the length of the diagonal from the midpoint?

 

[Leong]

You can find the slope of line segment AC (a diagonal) as well as its length and its midpoint. The negative reciprocal of that slope is the slope of the other diagonal since you're working with a square. Write an equation of a straight line using the point-slope form using the midpoint and the slope of the second diagonal.

let the equation to be the first equation.

find another circle equation that its center is at the midpoint of AC and the radius of half the length of AC. The circle will pass all the corner points of the square. solve the two equations will give the points of intersection of the circle and the straight line which will give point B and D.

(0,2) & (7,1).