Lesson 4

Students experiment with finding circles that circumscribe a quadrilateral. They observe that this appears to be possible for some quadrilaterals but not all of them. They’ll analyze the properties of quadrilaterals with circumscribed circles, or cyclic quadrilaterals, in the activities in this lesson.

Monitor for students who successfully circumscribe figure B.

Students may draw a curved shape that is not a circle to circumscribe figure C. Ask these students if their shape could be created by a compass.

Invite students to share their results. Ask which shape was the easiest (most likely figure A, the square). Display a successful result for figure B, and be sure students come to the conclusion that such a circle can’t be drawn for figure C.

Tell students that a circle is said to be circumscribed about a polygon if all the vertices of the polygon lie on the circle. If it is possible to draw a circumscribed circle for a quadrilateral, the figure is called a cyclic quadrilateral.

In this activity, students apply the Inscribed Angle Theorem to a series of problems with labeled angles. Then, they use the pattern they observe to draw a general conclusion about cyclic quadrilaterals.

Tell students that we’ve seen that some quadrilaterals have circumscribed circles but others don’t. We’ll look at a particular property of cyclic quadrilaterals, the ones that do have circumscribed circles.

If students don’t immediately recall the relationship between an inscribed angle and the arc it defines, suggest they look at their reference charts.

Invite students to share the value of \(\alpha + \beta\). Ask, “What word describes angle pairs whose measures add to 180 degrees?” (Supplementary.) Then, display the images from the warm-up for all to see.

A

B

C

Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Things students may notice:

• The first quadrilateral has 4 90 degree angles. It fits the idea that cyclic quadrilaterals have supplementary pairs of opposite angles.
• The third quadrilateral does not have supplementary pairs of opposite angles. One pair of opposite angles are both obtuse and the other pair are acute. This is why we couldn’t draw its circumscribed circle.

Things students may wonder:

• What are the angle measures in the second quadrilateral?
• We showed that cyclic quadrilaterals have supplementary pairs of opposite angles. Is the converse true? That is, if a quadrilateral has supplementary pairs of opposite angles, do we know for sure that it has a circumscribed circle?
• Do other shapes have circumscribed circles?

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the idea that the third quadrilateral does not have supplementary pairs of opposite angles does not come up, ask students to discuss this idea. Tell students that they will analyze circumscribed circles for triangles in a subsequent activity.

Students construct the circumscribed circle for a cyclic quadrilateral with a 90 degree angle. They observe that the diagonal connecting vertices adjacent to the one with a marked 90 degree angle must be a diameter of the circle, then find the diameter’s midpoint using construction tools or paper folding.

In the activity synthesis, students discuss the idea that the center of the circumscribed circle is equidistant from the vertices of the quadrilateral. This prepares students for a discussion of triangle circumcenters in a subsequent activity.

If students struggle to determine how diagonal \(BD\) relates to the circumscribed circle, remind them that \(BCD\) is an inscribed angle on the circle. Ask them what the right angle marking tells them about the measure of the arc going from \(B\) to \(D\) through \(A\).

The goal of the discussion is for students to notice that the center of the circumscribed circle is equidistant from all the vertices of the quadrilateral. This idea will be developed further in upcoming activities on triangle circumcenters.

Ask students how the distances \(AO, BO, CO,\) and \(DO\) compare (these distances are all the same because they all represent radii of the same circle). Then, display this image for all to see.

Ask students if this is a cyclic quadrilateral, and how they know. (Yes, it is a cyclic quadrilateral, because if we draw a circle with center \(A\) and radius \(AC\), it will go through all the other vertices since they’re the same distance from \(A\) as \(C\) is.)