Draw a diagram and think about the definition of a kite.
See solution.
Practice makes perfect
Let's put the given points P(0,0), Q(4,2), S(4,- 2), and H(4,0) on the diagram.
Since H and P are both on the x-axis and H is on diagonal PR, vertex R must also be on the x-axis to the right of H.
On the diagram, move point R to experiment with what quadrilateral PQRS looks like.
There are two conditions that quadrilateral PQRS must satisfy to be a kite.
It has to have two pairs of congruent consecutive sides. Since the x-axis is the perpendicular bisector of QS, this is always true if R is on the x-axis to the right of H.
Opposite sides must not be congruent. This is true if H is not the midpoint of PR, so R is not at (8,0). If R is at (8,0), then PQRS is a rhombus, which is not a kite.
To summarize, for R(r,0), quadrilateral PQRS is a kite if r>4 and r≠8.
Alternative Solution
Alternative way of thinking
The intersection of the diagonals can be understood in a less restrictive way.
We can think of H as the intersection point of the lines of the diagonals, not necessarily the segments.
Using this broader understanding, there are other possibilities for the position of R.
In this case R can be anywhere on the x-axis except at points (0,0), (4,0), and (8,0).
If R is at (0,0), then PQRS is not a quadrilateral.
If R is at (4,0), then PQRS is a triangle.
If R is at (8,0), then PQRS is a rhombus, not a kite.
