Place the diagonals of the kite along the axes of the coordinate plane.
See solution.
Practice makes perfect
We are asked to write a coordinate proof about a kite.
Recall that the diagonals of a kite are perpendicular and one diagonal bisects the other.
We can place the kite in the coordinate plane so that the diagonals are placed along the axes and the y-coordinates of the vertices on the y-axis are opposites of each other.
Since we are asked to investigate the quadrilateral formed by the midpoints of the sides, let's use 2a, 2b, and 2c as the coordinates of the vertices instead of a, b, and c. This way, we will not have fractions.
Let's use the Midpoint Formula to find the coordinates of the midpoints.
Endpoints
Midpoint (x_1+x_2/2,y_1+y_2/2)
Substitution
Simplification
A(2a,0) and B(0,2b)
(2a+0/2,0+2b/2)
E(a,b)
B(0,2b) and C(2c,0)
(0+2c/2,2b+0/2)
F(c,b)
C(2c,0) and D(0,- 2b)
(2c+0/2,0+(- 2b)/2)
G(c,- b)
D(0,- 2b) and A(2a,0)
(0+2a/2,- 2b+0/2)
H(a,- b)
Let's add these coordinates to the diagram.
Let's investigate the sides of quadrilateral EFGH.
Claim
Justification
EF is horizontal.
The y-coordinates of E(a,b) and F(c,b) are the same.
FG is vertical.
The x-coordinates of F(c,b) and G(c,- b) are the same.
GH is horizontal.
The y-coordinates of G(c,- b) and H(a,- b) are the same.
HE is vertical.
The x-coordinates of H(a,- b) and E(a,b) are the same.
Let's see what these observations imply.
Opposite sides are parallel, so quadrilateral EFGH is a parallelogram.
Horizontal and vertical lines are perpendicular, so the angles of this parallelogram are right angles. This means that EFGH is a rectangle.
This completes the proof that the quadrilateral formed by connecting the midpoints of a kite is a rectangle.
