SOLUTION: Hello, I have 4 points given to me and I have to prove algebraically whether the figure is a kite. I can't just plot the points. How would I go about doing that?

Algebra ->  Geometry-proofs -> SOLUTION: Hello, I have 4 points given to me and I have to prove algebraically whether the figure is a kite. I can't just plot the points. How would I go about doing that?      Log On


   

Question 297859: Hello,
I have 4 points given to me and I have to prove algebraically whether the figure is a kite. I can't just plot the points. How would I go about doing that?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Given 4 points, divide them into two sets.
One set is the lengthwise component on the kite (line AB below).
The other set is the transverse (sideways) component of the kite (line CD).

Calculate the eqaution of the line using the slope and the point-slope form of a line for the length wise component.
y-ya=m1%28x-xa%29
m1=%28yb-ya%29%2F%28xb-xa%29
Then convert to slope-intercept form
y=m1%2Ax%2Bb1
Do the same for the transverse component to get its equation,
y=m2%2Ax%2Bb2
If it's a kite, the lines will be perpendicular and the relationship between the two slopes will be,
m1%2Am2=-1
If that's not the case, then stop there. It's not a kite.
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If it is true then find the intersection point, I, between the two lines.
Solve the system of equations.
y=m1%2Ax%2Bb1
y=m2%2Ax%2Bb2
Since they both equal y, set them equal to each other.
m1%2Axi%2Bb1=m1%2Axi%2Bb2
Solve for x.
Then solve for y using either equation.
Now you have the point of intersection (xi,yi) of the two components of the kite.
If it's a kite, the distance from the intersection point (xi,yi) to the ends of the transverse components (C and D) must be identical.
Dci=Ddi
Dci%5E2=Ddi%5E2
%28xc-xi%29%5E2%2B%28yc-yi%29%5E2=%28xd-xi%29%5E2%2B%28yd-yi%29%5E2
If that's the case, then you've proven that the points form a kite.
If it's not true, then you don't have a kite.