Question

point o is the center of the circle. what is the perimeter of quadrilateral dobc? 28 units 16 units 22 units 14 units

Explanation:

Step1: Recall circle - tangent properties

Since $CD$ and $CB$ are tangents to the circle with center $O$, $OD\perp CD$ and $OB\perp CB$, and $OD = OB$ (radii of the same circle), and $OC$ is common. So, right - triangles $ODC$ and $OBC$ are congruent by the Hypotenuse - Leg (HL) congruence criterion. Then $CD=CB = 8$ and $OD = OB$. Let $OD = OB=r$. In right - triangle $ODC$, using the Pythagorean theorem, if $OD = r$ and $CD = 8$ and $OC$ is the hypotenuse. Also, in right - triangle $OBC$, $OB = r$ and $CB = 8$.

Step2: Identify side lengths of the quadrilateral

In quadrilateral $DOBC$, $DO = OB$ (radii), $CD = CB=8$, and assume $DO = OB = 6$ (from the figure, if we consider the right - triangle formed with the radius and the tangent). The sides of the quadrilateral $DOBC$ are $DO = 6$, $OB = 6$, $BC = 8$, $CD = 8$.

Step3: Calculate the perimeter

The perimeter $P$ of a quadrilateral is the sum of its side lengths. $P=DO + OB+BC + CD$. Substitute $DO = 6$, $OB = 6$, $BC = 8$, $CD = 8$ into the formula. $P=6 + 6+8 + 8=28$.

Answer:

28 units