I have no idea how to solve this problem. The question says:

Determine whether the points A(4,5), B(-3,3), C(-6,-13) and D(6,-2) are the vertices of a kite. Explain your answer.
Can you please help me with the steps to solve it and understanding it.

Graph all the points given on the xy-plane. If you get a picture of a KITE, then the answer is yes.

Of course, I graphed all the points for you and determined that the given 4 points form a picture of a kite.

Done!

first draw it.

It looks kind of like a kite with the tip at A, the tail at C, the wing tips at B and D
let's see if AC is perpendicular to BD
slope C to A = (5+13)/(4+6) = 18/10 = 1.8
slope B to D = (-2-3)/(6+3) = -5/9
ah ha, slope 2 = -1/slope 1, they are perpendicular
This sure looks like a kite.
the final proof would be to find the intersection P of the lines BD and AC and see if BP = PD

Thanks! =)

But one more question about this...
I looked in the back of the book for help because that's where all the answers are (No I don't use it to cheat. Knowing the answers helps me to figure how the steps to solve it.) and it said something about slopes and using the distance formula. Am I suppose to use that instead?

geometric concept is modeled by the beam of a flashlight

Sure! To determine whether the given points A(4,5), B(-3,3), C(-6,-13), and D(6,-2) are the vertices of a kite, we need to follow a few steps.

Step 1: Calculate the distances between each pair of points.
First, calculate the distances between points A and B, A and C, A and D, B and C, B and D, and C and D. Use the distance formula:

Distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2-x1)^2 + (y2-y1)^2)

For example,
Distance between A(4,5) and B(-3,3) is: d_AB = √((-3-4)^2 + (3-5)^2) = √((-7)^2 + (-2)^2) = √(49 + 4) = √53

Step 2: Check the conditions for a kite.
To determine if the points form a kite, we need to check the following conditions:
a) All four sides must have equal length.
b) The diagonals must be perpendicular to each other.
c) The diagonals must bisect each other.

Step 3: Verify the conditions.
a) Check if all four sides have equal length. Compare the distances calculated in Step 1.
- If all four sides have the same length, go to Step 3b.
- If any of the four sides have different lengths, then the points are not the vertices of a kite, and you can conclude that they do not form a kite.

b) Check if the diagonals are perpendicular to each other.
- Find the slopes of the lines AB and CD. If the product of their slopes is -1, then the diagonals are perpendicular.
Slope of AB = (y2-y1)/(x2-x1) = (3-5)/(-3-4) = -2/-7 = 2/7
Slope of CD = (y2-y1)/(x2-x1) = (-2-(-13))/(6-(-6)) = 11/12
The product of the slopes is: 2/7 * 11/12 = 22/84 = 11/42
Since the product of slopes is not -1, the diagonals are not perpendicular, and the points do not form a kite.

c) Check if the diagonals bisect each other.
- Find the midpoint of AC and BD. If the midpoints are the same, then the diagonals bisect each other.
Midpoint of AC = ((x1+x3)/2, (y1+y3)/2) = ((4+(-6))/2, (5+(-13))/2) = (-1, -4)
Midpoint of BD = ((x2+x4)/2, (y2+y4)/2) = ((-3+6)/2, (3+(-2))/2) = (1.5, 0.5)
Since the midpoints are different, the diagonals do not bisect each other, and the points do not form a kite.

Based on the calculations in Step 3, we can conclude that the given points A(4,5), B(-3,3), C(-6,-13), and D(6,-2) do not form the vertices of a kite.