-Use the Drawing Provided to Explain the Following Theorem

Question

Examlex · Accepted Answer

The Answer is To "box" the quadrilateral

, we draw auxiliary lines as follows:
through point D, we draw

; through point B, we draw

;
through point A, we draw

; and through point C, we draw

.
The quadrilateral formed is a parallelogram that can be shown to have a right angle;
this follows from the fact that

is a parallelogram that contains a right angle at
vertex F . . . so the opposite angle (at vertex R) must also be a right angle.
Because

is a diagonal of

(actually rectangle

,

;
that is, a diagonal of a parallelogram separates the parallelogram into 2 congruent

.
Similarly,

,

, and

. Thus, the area
of quadrilateral

is one half of that of rectangle

.
But the area of

is

, so

is given by

.

Definitions: