Question

The long diagonal of a kite is a perpendicular bisector to the short diagonal. Using this piece of information, how can you prove that adjacent sides are congruent in a kite?

Answer 1

@nailpolisheverywhere @OOOPS @aajugdar @Nurali

Answer 2

Label the four vertices of the kite as A, B, C, D going in clockwise direction, for example.

Now we have two diagonals AC, and BD. Assuming AC is the long diagonal it intersects and
bisects BD at a point, E , for example.

Consider triangles AEB, and AED, angle AEB = angle AED = 90, because the two diagonals are perpendicular. And EB = ED because AC bisects BD. And third, AE is common between the two triangles.

Therefore, the two triangles are congruent by side-angle-side.
It follows that AB is congruent with AD. Thus the two adjacent sides are congruent.

In a similar fashion, we can show that CB and CD are congruent.

Using the results above we have also shown that angle BAE = DAE , therefore diagonal AC
bisects angle DAB. And also, we have shown that angle BCE = DCE, therefore diagonal AC
bisects angle DCB.

Answer 3

this will help u get it

Answer 4

perpendicular bisector divides a line in two equal parts,so DE =EB
and 90* angle to both
and a common side AE
so by side angle side property the triangles are congruent
so the adjacent sides AD and DB are congruent :)