Question

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?

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

I don't understand how to even lay out the problem... Thats why its so hard because they also want you to explain.

Answer 1

`ABCD` is a kite.
The diagonals are perpendicular and the long diagonal bisects the short diagonal.

Explanation:

You need to know the properties of a kite and prove that they are true for the given points, There are several options:

• `2` pairs of adjacent sides are equal. Sketch the points to get an idea of which sides might be equal. Use the distance formula to prove that `AB = AD and BC = DC`
  This is a long and repetitive method, but easy to do.

• one pair of opposite angles are equal. NOt easy to find the angles from the vertices ... skip this method.

• The diagonals intersect at `90°` and one diagonal bisects the other. Find the slope of each diagonal, then check whether one diagonal bisects the other.

`m_(AC) = (5-(-13))/(4-(-6)) = 18/10 = 9/5`

`m_(BD) = (3-(-2))/(-3-6) = 5/(-9) = -5/9`

`9/5 xx-5/9 =-1" ":. AC and BD` are perpendicular

Is `AD` a line of symmetry?
If so it will pass through the midpoint of `BC`

`M_(BD) = ((-3+6)/2; (3+(-2))/2) = M(1 1/2, 1/2)`

Equation of `AD`: m= 9/5 and (4,5)#

`y-y_1 = m(x-x_1)" "rarr y-5=9/5(x-4)`

`y = 9/5x-36/5+5" "rarr y= 9/5x-2 1/5`

Is `M` on this line? Substitute to check..

`1/2 = 9/5(3/2)-2 1/5`

`1/2 = 1/2`

`ABCD` is a kite.
The diagonals are perpendicular and the long diagonal bisects the short diagonal.