Exercises: Inscribed and Circumscribed Circles

The perpendicular bisector of a segment is the set of all points that are equidistant from the segment's two   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

An angle bisector of an angle is the ray that divides the angle into two equal parts. A point on the angle bisector is equidistant from the two   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   of the angle.

The inscribed circle (incircle) of a triangle is the circle that   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

The circumcenter of an obtuse triangle lies   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Triangle $ABC$ is inscribed in a circle with center $O$. If $OA = 9$ cm, what is the length $OC$? Give your answer in centimeters.

To locate the incenter of triangle $PQR$, you construct   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Triangle $XYZ$ has an incircle with center $I$. The inradius (the perpendicular distance from $I$ to side $\overline{XY}$) is 5 cm. What is the perpendicular distance from $I$ to side $\overline{YZ}$? Give your answer in centimeters.

Cyclic quadrilateral $ABCD$ is inscribed in a circle. If $\angle A = 72\degree$, what is $\angle C$?

Quadrilateral $ABCD$ is inscribed in a circle. The angle measures are $\angle A = 85\degree$, $\angle B = 95\degree$, $\angle C = 95\degree$, and $\angle D = 85\degree$. Which statement correctly describes these angles?

To prove the three perpendicular bisectors of triangle $ABC$ are concurrent, we argue: Let $O$ be the intersection of the perpendicular bisectors of $\overline{AB}$ and $\overline{BC}$. Since $O$ is on the perpendicular bisector of $\overline{AB}$, we know $OA = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$. Since $O$ is on the perpendicular bisector of $\overline{BC}$, we know $OB = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$. Therefore $OA = OC$ by   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   , so $O$ also lies on the perpendicular bisector of $\overline{AC}$.

Triangle $RST$ has an incircle with center $I$ and a circumscribed circle with center $O$. Which statement is always true?

Explain in your own words why you only need to construct two angle bisectors to find the incenter of a triangle. Why is the third angle bisector guaranteed to pass through the same point?

Quadrilateral $EFGH$ has $\angle E = 80\degree$, $\angle F = 105\degree$, $\angle G = 100\degree$, $\angle H = 75\degree$. Can quadrilateral $EFGH$ be inscribed in a circle?

What is the circumradius (the radius of the fountain) in meters?

Find $\angle D$ in degrees.

What is the error in Mia's reasoning?

Identify all errors in Leo's statement.

Write a proof that the three perpendicular bisectors of the sides of triangle $ABC$ are concurrent (all meet at one point). Your proof should explain why two perpendicular bisectors are sufficient to find the circumcenter, and why the third is guaranteed to pass through the same point.

Write a proof that opposite angles of any cyclic quadrilateral are supplementary. Let $ABCD$ be inscribed in a circle. Your proof should use the Inscribed Angle Theorem to show $\angle A + \angle C = 180\degree$.