Properties of Quads - Kite diagonals
This applet contains hints on using triangle congruence to show that one diagonal of a kite always bisects the other and that the diagonals are perpendicular to each other.
An alternative justification is hinted after the applet.
Alternative justification
If we assume that we have already proven that the set of all points equidistant from two points X, Y in the plane is the perpendicular bisector of the line segment XY (and this is very reasonable assumption), we can use it to justify the properties of kite diagonals. Labels refer to the picture in the applet above.
- We know that point A is equidistant from BD and point C is also equidistant from BD.
- They must therefore lie on the perpendicular bisector of BD.
- But there is exactly one line going through two distinct points, so the line AC must be the perpendicular bisector of BD.
- This explains why AC bisects BD and why AC BD.