We have been told that if every side of a polygon is tangent to a circle, then the polygon is said to be circumscribed about the circle.
In the above diagram, we know that quadrilateral ABCD is circumscribed about ⊙ Q. We will determine whether the following statement is always true.
AB+CD=AD+BC
Tangent segments from a common external point are congruent.
Using this theorem, we can identify the congruent tangent segments on the diagram.
As we can see, four pairs of segments are congruent. Note that congruent segments have the same lengths.
AW &= AZ
WB&=BX
CY&=XC
YD&=ZD
Now, let's add the left-hand and right-hand sides of the above equations.
AW+WB +CY+YD
=
AZ+ BX + XC+ZD
Each segment is a subsegment of either AB, CD, AD, or BC. With this information, we will regroup one side of the obtained equation by using the Commutative Property of Addition. Then, we can add the subsegments to apply the Segment Addition Postulate.
AW+WB^(AB) +CY+YD^(CD)
=
AZ+ZD_(AD)+ BX + XC_(BC)
⇕
AB+CD=AD+BC
Therefore, the statement is always true.
