Students experiment with finding circles that circumscribe a quadrilateral. They observe that this appears to be possible for only some quadrilaterals. 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 Quadrilateral B.
Launch
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Activity
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Student Task Statement
For each quadrilateral, use a compass to see if you can draw a circle that passes through all 4 of the quadrilateral’s vertices.
A
B
C
Student Response
Activity Synthesis
The goal is for students to share their ideas with the group before new vocabulary is made explicit.
Invite students to share their results. Ask which shape was the easiest to draw a circle around (most likely Quadrilateral A, the square). Display a successful result for Quadrilateral B, and be sure students come to the conclusion that such a circle can’t be drawn for Quadrilateral 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 quadrilateral is called a cyclic quadrilateral.
The goal is to continue to think about the center of the circumscribed circle of a cyclic quadrilateral as being equidistant from the vertices of the figure. Display this image for all to see.
Tell students that each vertex of the quadrilateral represents a town. A construction company says it is planning to build a shopping mall the same distance from each town. Clare claims this is impossible. How does she know? Give students 1 minute of quiet work time and then 1 minute to share their work with a partner. Follow with a whole-class discussion.
Sample response: If there were a point that was the same distance from all four town centers, then it would be possible to draw a circle centered at this point that passed through all 4 vertices. That is, the towns would form a cyclic quadrilateral. In that case, the opposite angles of the quadrilateral must be supplementary. However, the opposite angles of our quadrilateral are not supplementary, so it can’t be cyclic.
Student Lesson Summary
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. Not all quadrilaterals have this property.
We can prove that the opposite angles of a cyclic quadrilateral are supplementary. Consider opposite angles and , labeled and .
Angle is inscribed in the arc from to through . Angle is inscribed in the arc from to through . Together, the two arcs trace out the entire circumference of the circle, so their measures add to 360 degrees. By the Inscribed Angle Theorem, the sum of and must be half of 360 degrees, or 180 degrees. So angles and are supplementary. The same argument can be applied to the other pair of opposite angles.
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Students may draw a curved shape that is not a circle to circumscribe Quadrilateral C. Ask these students if their shape could be created by a compass.
In this activity, students apply the Inscribed Angle Theorem to a series of problems with labeled angles. Then they use the pattern they observe from repeated calculations to draw a general conclusion about cyclic quadrilaterals (MP8).
Launch
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 quadrilaterals that do have circumscribed circles.
Activity
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Student Task Statement
The images show 3 quadrilaterals with circumscribed circles.
A
B
C
For each one, highlight the arc from to passing through . Then find the measures of:
the arc you highlighted
the other arc from to
angle
Here is another quadrilateral with a circumscribed circle. What is the value of ? Explain or show your reasoning.
Student Response
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Building on Student Thinking
If students don’t immediately recall the relationship between an inscribed angle and the arc it defines, suggest they look at their reference charts.
Activity Synthesis
The goal of this discussion is for students to consider which types of quadrilaterals have supplementary pairs of opposite angles. Invite students to share the value of . Ask, “What word describes angle pairs whose measures add to 180 degrees?” (“supplementary”)
Tell students to close their books or devices (or to keep them closed). Then display the images from the Warm-up for all to see:
A
B
C
Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the images.
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.
Things students may notice:
The first quadrilateral has four 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 is made of two obtuse angles, and the other pair is made of two acute angles. 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?
MLR8 Discussion Supports. To support the transfer of new vocabulary to long-term memory, invite students to chorally repeat these words or phrases 1–2 times: “quadrilateral,” “cyclic quadrilateral,” “circumscribed circle,” “supplementary angles,” and “inscribed angle.” Advances: Speaking, Conversing
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in students’ thinking about the value of . For example, color code the inscribed angles and the two arcs that create the complete circle. Use the examples that demonstrate that the angles add to 180 degrees, and point out that if one angle is made bigger, the other gets smaller to bridge to the abstract rule. Supports accessibility for: Visual-Spatial Processing
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, and then they find the diameter’s midpoint using construction tools or paper folding.
In the digital version of the activity, students use an applet to construct the circumscribed circle for a quadrilateral. The applet allows students to use two of the quadrilateral’s endpoints to draw the diameter of the circle and then find its center. Use the digital version if students will benefit from seeing the relationship in a dynamic way.
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.
Launch
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation:
The goal of the discussion is for students to understand 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 and 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 and radius , it will go through all the other vertices since they’re the same distance from as is.)
Standards Alignment
Building On
Addressing
HSG-C.A.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Draw diagonal . How will this diagonal relate to the circumscribed circle? Explain your reasoning.
Construct the center of the circumscribed circle for quadrilateral . Label this point . Explain why your method worked.
Construct the circumscribed circle for quadrilateral .
Could we follow this procedure to construct a circumscribed circle for any cyclic quadrilateral? Explain your reasoning.
Student Response
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Building on Student Thinking
If students struggle to determine how diagonal relates to the circumscribed circle, remind them that is an inscribed angle on the circle. Ask them what the right angle marking tells them about the measure of the arc going from to through .
