How to Determine a Kite

Determine if quadrilateral ABCD is a kite. A kite is a quadrilateral with two pairs of adjacent sides, congruent. A kite also has perpendicular diagonals, where one bisects the other. You can use either of these things to determine if a quadrilateral is a kite. I’m going to use the first method to determine if this quadrilateral, ABCD, is a kite.

That means that I need to show that both pairs of my adjacent sides are congruent, or that I have two pairs of adjacent sides that are congruent. In order to tell if they’re congruent, I’m going to find the distance between the points, and if the distances between the points are the same, then that means that those segments are congruent.

We’re going to use the distance formula, which is distance equals the square root of x_2 minus x_1, squared, plus, y_2 minus y_1, squared, (and I’ve already put the labels on all of my points) and it doesn’t matter which point you call x_1, y_1, or x_2, y_2, as long as you’re consistent.

In one point, like point A, you couldn’t call one coordinate x_1 and then the other one y_2, so as long as you don’t mix it up like that, you can label any of the points x_1, y_1, x_2, y_2, it doesn’t matter. First, I want to find the distance from A to B, so that’s the square root of x_2 is 1, minus x_1 is negative 2, squared, plus, y_2 is 3, minus y_1 is 2, squared.

Now we need to simplify underneath the radical, starting with the parentheses, so that equals the square root of, (can add the inverse) 1 plus 2 is 3, squared, plus, 3 minus 2 is 1, squared, that equals the square root of 3 squared is 9, plus, 1 squared is 1, so that equals the square root of 10.

The square root of 10 is an irrational number meaning that it goes on forever, its decimal does not end, but it’s not really important what the distance is, what’s important is if my distances are equal. I’m not actually going to solve for the square root of 10 and find the actual square root of the number, the square root of 10 is good enough for our purposes.

Now I’m going to find the distance between points B and C. I’m using the distance formula again, so the square root of x_2, (using B and C) so x_2 is 1, minus x_1 is 2, squared, plus, y_2 is 3, minus y_1 is 0, squared. Again, we’re going to simplify underneath the radical first, starting with our parentheses.

1 minus 2, or plus a negative 2, is negative 1 squared, plus, 3 minus 0 is 3 squared. Negative 1 squared, is 1, plus, 3 squared is 9. The square root of 1 plus 9 is the square root of 10, and that’s the same thing that we got for the distance from A to B, so that means the distances from A to B and B to C are equal, and since the distances are equal that means those segments are congruent, so I’m going to mark those congruent.

Now if I can prove that the distance—from A to D, and from D to C—if I can prove that those distances are the same, that means that those segments are congruent, and I’ll have proved that this quadrilateral is a kite. We’re going to use the distance formula, again, to find the distance from C to D (and that’s a C).

Now using C and D, (x_1, y_1, x_2, y_2) and it’s equal to the square root of (same formula) x_2, x_2 is negative 2, minus x_1 is 2, squared, plus, y_2 minus y_1, y_2 is negative 3, minus y_1 is 0, squared, and we need to simplify under our radical, starting with our parentheses. That equals the square root of (add the inverse) negative 2 plus negative 2 is negative 4, negative 4 squared, plus, negative 3 minus 0 is negative 3, squared.

I need to keep simplifying, negative 4 squared is positive 16, (a negative times a negative is a positive) plus, negative 3 squared is positive 9, and that’s equal to the square root of 25, 16 plus 9 is 25, and the square root of 25 is 5. Finally, I need to find the distance from A to D, and that’s equal to the square root of (same formula again) x_2 minus x_1, so my x_2 for A and D is negative 2, minus x_1 is negative 2, squared, plus, y_2 minus y_1 squared, y_2 is negative 3, minus y_1 is 2, squared, and that equals the square root of (add the inverse) negative 2 plus 2 is 0, 0 squared, plus, (add the inverse) negative 3 plus negative 2 is negative 5, squared.

That equals the square root of 0 squared is 0, plus, negative 5 squared is 25, so it’s the square root of 25, which is also 5. Since the distance from C to D is equal to the distance from A to D, that means that segment AD is congruent to segment CD, which means that this quadrilateral is, in fact, a kite. The other method for determining if this quadrilateral is a kite, is to find the slopes of the diagonals of the kite, and if the slopes of the diagonals of the kite are opposite reciprocals, that means that those lines are perpendicular.

Then find the midpoint of each one of the diagonals, and if one of your segments bisects the other one or goes through the mid-point of that segment, then that means that it is a kite, so that’s another way to do it. Or you can find the distances between your points, to find if you have two pairs of adjacent sides that are congruent.