SOLUTION: Prove that the midpoint of a kite form a rectangle

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Question 1091561: Prove that the midpoint of a kite form a rectangle
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
HINT.  Below find the skeleton  (the scheme)  of the proof. 

1.  Use the fact that the mid-line of a triangle 

        a) is parallel to the base of the triangle and 

        b) has the length half of that of the base.


           See the lesson  The line segment joining the midpoints of two sides of a triangle  in this site.  


    In this way, you prove that the quadrilateral formed by the side midpoints of the kite has opposite sides parallel and congruent.

    It implies that this quadrilateral is a parallelogram.



2.  Now you need to prove that the diagonals of a kite are perpendicular.

    Draw these diagonals.

         a) notice that one diagonal divides the kite in two congruent triangles
            (according to SSS-congruency test for triangles).

            Hence, the common side, which is the diagonal, is at the same time the bisector of the two kite's angles.


         b) From the other side, the other diagonal cuts the kite in two isosceles triangles.

            Since the first diagonal is the angle bisector of these triangles, it is the altitude at the same time.

            Thus we proved that the diagonals of a kite are perpendicular.


3.  It implies that the parallelogram under the question is a rectangle.

Proved.


Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

This can be proved rather easily using coordinate geometry. Suppose we situate the kite on the coordinate plane so that the intersection of the two diagonals is the origin. According to the definition of a kite, we can call two of the vertices (-2a,0) and (2a,0), and the other two vertices (0,2b) and (0,2c) (where in my diagram b is positive and c is negative).

Then the midpoints of the four sides are (-a,b), (a,b), (-a,d), and (a,d).

The quadrilateral determined by those four points has two sides that are horizontal line segments and two sides that are vertical line segments; that makes the quadrilateral a rectangle.
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