# Lesson 4

## 4.1: Connecting the Dots (5 minutes)

### Warm-up

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.

### Student Facing

For each quadrilateral, use a compass to see if you can draw a circle that passes through all 4 of the quadrilateral’s vertices.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

## 4.2: Inscribed Angles and Circumscribed Circles (20 minutes)

### Activity

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.

### 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 ones that do have circumscribed circles.

### Student Facing

1. The images show 3 quadrilaterals with circumscribed circles.
For each one, highlight the arc from $$S$$ to $$Q$$ passing through $$P$$. Then, find the measures of:
1. the arc you highlighted
2. the other arc from $$S$$ to $$Q$$
3. angle $$SPQ$$
2. Here is another quadrilateral with a circumscribed circle. What is the value of $$\alpha + \beta$$? Explain or show your reasoning. ### Student Facing

#### Are you ready for more?

Brahmagupta’s formula states that for a quadrilateral whose vertices all lie on the same circle, the area of the quadrilateral is $$\sqrt{(s-a)(s-b)(s-c)(s-d)}$$ where $$a,b,c,$$ and $$d$$ are the lengths of the quadrilateral’s sides and $$s$$ is half its perimeter.

In the cyclic quadrilateral in the image, point $$O$$ is the center of the quadrilateral’s circumscribed circle. Validate Brahmagupta’s formula for this particular quadrilateral by first finding the sum of the areas of the top and bottom triangles. Then, calculate the area again using Brahmagupta’s formula.

### Anticipated Misconceptions

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

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.

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.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking about the value of $$\alpha+\beta$$. Highlight 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; Conceptual processing
Speaking: MLR8 Discussion Supports. Use this to routine amplify students' use of mathematical language. Encourage precise use of mathematical language when students share the things they noticed and wondered. Amplify words and phrases such as, quadrilateral, cyclic quadrilateral, circumscribed circle, supplementary angles, and inscribed angles. Invite students to chorally repeat phrases that include these words in context. Choral repetition facilitates the transfer of vocabulary to students’ long term memory.
Design Principle(s): Optimize output (for explanation)

## 4.3: Construction Ahead (10 minutes)

### 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.

### 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 such as: “If _____ then _____ because. . .”, “_____ reminds me of _____ because . . .”,  or “Is it always true that . . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Quadrilateral $$ABCD$$ is a cyclic quadrilateral.

1. Draw diagonal $$BD$$. How will this diagonal relate to the circumscribed circle? Explain your reasoning.
2. Construct the center of the circumscribed circle for quadrilateral $$ABCD$$. Label this point $$O$$. Explain why your method worked.
3. Construct the circumscribed circle for quadrilateral $$ABCD$$.
4. Could we follow this procedure to construct a circumscribed circle for any cyclic quadrilaterals? Explain your reasoning.

### 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 such as: “If _____ then _____ because. . .”, “_____ reminds me of _____ because . . .”,  or “Is it always true that . . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Quadrilateral $$ABCD$$ is a cyclic quadrilateral.

1. Draw diagonal $$BD$$. How will this diagonal relate to the circumscribed circle? Explain your reasoning.
2. Construct the center of the circumscribed circle for quadrilateral $$ABCD$$. Label this point $$O$$. Explain why your method worked.
3. Construct the circumscribed circle for quadrilateral $$ABCD$$.
4. Could we follow this procedure to construct a circumscribed circle for any cyclic quadrilaterals? Explain your reasoning.

### Anticipated Misconceptions

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 $$A$$.

### Activity Synthesis

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.)

## Lesson Synthesis

### Lesson Synthesis

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 are not supplementary, so it can’t be cyclic.

## Student Lesson Summary

### Student Facing

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 $$BCD$$ and $$BAD$$, labeled $$\alpha$$ and $$\beta$$.
Angle $$BAD$$ is inscribed in the arc from $$B$$ to $$D$$ through $$C$$. Angle $$BCD$$ is inscribed in the arc from $$B$$ to $$D$$ through $$A$$. Together, the 2 arcs trace out the entire circumference of the circle, so their measures add to 360 degrees. By the Inscribed Angle Theorem, the sum of $$\alpha$$ and $$\beta$$ must be half of 360 degrees, or 180 degrees. So angles $$BAD$$ and $$BCD$$ are supplementary. The same argument can be applied to the other pair of opposite angles.