Question

Rectangle KLMN has vertices `K(0,4), L(4,2), M(1,-4)`, and `N(-3,-2)`. What are the coordinates of the point of intersection of the diagonals?

Answer 1

`(.5,0)`

Explanation:

To answer the question we need to find the equations of the diagonals KM and LN and combine them to find the point or intersection.

The formula for the equation of the line is
`(y-y_0)=k(x-x0)`
where
`k=(Delta y)/(Delta x)=(y_1-y_0)/(x_1-x_0)`

Equation of the line 1 in which the diagonal KM lays:
`k_(KM)=(-4-4)/(1-0)=-8`
`(y-4)=-8x` => `y=-8x+4`

Equation of line 2 in which the diagonal LN lays:
`k_(LN)=(-2-2)/(-3-4)=4/7`
`(y-2)=(4/7)(x-4)` => `y=(4x-16)/7+2` => `y=(4x-2)/7`

Combining the equations of lines 1 and 2:
`-8x+4=(4x-2)/7` => `-56x+28=4x-2` => `60x=30` => `x=1/2`
=> `y=-8(1/2)+4=-4+4` => `y=0`

So the point in which the rectangle's diagonals meet is `(1/2,0)`