Lesson 5

Triangles in Circles

5.1: One Perpendicular Bisector (5 minutes)

Warm-up

In this activity, students recall that points on perpendicular bisectors of segments are the same distance from each endpoint of the segment. In the next activity, students will construct the remaining perpendicular bisectors of this triangle and then construct the triangle’s circumscribed circle.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Student Facing

The image shows a triangle.

1. Construct the perpendicular bisector of segment $$AB$$.
2. Imagine a point $$D$$ placed anywhere on the perpendicular bisector you constructed. How would the distance from $$D$$ to $$A$$ compare to the distance from $$D$$ to $$B$$? Explain your reasoning.

Activity Synthesis

Display this image for all to see. Tell students that the dashed line is the perpendicular bisector of segment $$AB$$.

For each of the points $$E$$, $$F$$, and $$G$$, ask students if the point is closer to $$A$$ or closer to $$B$$, and ask how they know. (Point $$F$$ is closer to $$A$$. Point $$G$$ is closer to $$B$$. Point $$E$$ is the same distance from $$A$$ as it is to $$B$$. All points to the left of the perpendicular bisector are closer to $$A$$, points to the right are closer to $$B$$, and points on the bisector are equidistant from $$A$$ and $$B$$.)

5.2: Three Perpendicular Bisectors (20 minutes)

Activity

Students construct the circumcenter of the triangle from the previous activity, or the point where the perpendicular bisectors of the sides of the triangle intersect. They determine that the circumcenter is equidistant from the vertices of the triangle based on the properties of perpendicular bisectors. Finally, they construct the circumscribed circle, observing that this is possible because of the equal distances already noted.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Student Facing

1. Construct the perpendicular bisector of segment $$BC$$ from the earlier activity. Label the point where the 2 perpendicular bisectors intersect as $$P$$.
2. Use a colored pencil to draw segments $$PA,PB,$$ and $$PC$$. How do the lengths of these segments compare? Explain your reasoning.
3. Imagine the perpendicular bisector of segment $$AC$$. Will it pass through point $$P$$? Explain your reasoning.
4. Construct the perpendicular bisector of segment $$AC$$.
5. Construct a circle centered at $$P$$ with radius $$PA$$.
6. Why does the circle also pass through points $$B$$ and $$C$$?

Student Facing

Points $$A,B,$$ and $$C$$ are graphed. Find the coordinates of the circumcenter and the radius of the circumscribed circle for triangle $$ABC$$.

Anticipated Misconceptions

Due to the precision of the tools, students’ circles may not pass through all 3 vertices. Explain that if we had more precise tools, the distances would be the same and the circle would pass through all the vertices.

Activity Synthesis

The goal is to conclude that all triangles have a circumscribed circle.

• “What kind of circle did you construct in relation to the triangle?” (It is a circumscribed circle.)
• “We saw that some quadrilaterals have a circumscribed circle, but some do not. Do all triangles have a circumscribed circle, or only some?” (All triangles have a circumscribed circle, because what we did for triangle $$ABC$$ could be done for any triangle. We used properties of perpendicular bisectors, not anything special about this triangle.)
• “What other special lines in triangles have we seen that all meet at a central point?” (In the Coordinate Geometry unit, we saw that the medians and the altitudes in a triangle also have single points of intersection.)

Tell students that the point where all 3 perpendicular bisectors of a triangle intersect is called the triangle’s circumcenter. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

The 3 perpendicular bisectors of the sides of a triangle meet at a single point, called the triangle’s circumcenter. This point is the center of the triangle’s circumscribed circle. (Theorem)

If time permits, ask: “For the triangle, once we drew 2 perpendicular bisectors, we knew the third would pass through the point of intersection of the first 2. Why doesn’t a similar argument work for a quadrilateral? Use a general quadrilateral $$ABCD$$ as an example.” (Draw the perpendicular bisectors of segments $$AB$$ and $$BC$$ and call their point of intersection $$P$$. This point is equidistant from points $$A,B,$$ and $$C$$. In a triangle, there are only 3 vertices, so that covers all of them. However, now we have 4 vertices. We have no way to guarantee that point $$P$$ is equidistant from point $$D$$.)

5.3: Wandering Centers (10 minutes)

Activity

In this activity, students compare the circumcenters of acute, obtuse, and right triangles. They find that the circumcenter of a right triangle lies on its hypotenuse, the circumcenter of an acute triangle lies inside the triangle, and the circumcenter of an obtuse triangle lies outside the triangle.

This activity works best when each student has access to devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way.

Launch

Arrange students in groups of 2.

Engagement: Develop Effort and Persistence. Provide background knowledge to help focus on increasing the length of on-task orientation in the face of distractions. For example, provide images and definitions of obtuse, acute, and right triangles.
Supports accessibility for: Attention; Social-emotional skills

Student Facing

Move the vertices of triangle $$ABC$$ and observe the resulting location of the triangle’s circumcenter, point $$D$$. Determine what seems to be true when the circumcenter is in each of these locations:

1. outside the triangle
2. on one of the triangle’s sides
3. inside the triangle

Launch

Arrange students in groups of 3–4. Ask students, “Suppose you construct 2 perpendicular bisectors of a triangle. Do you need to construct the third to find the circumcenter? Why or why not?” (We don’t need to construct the third perpendicular bisector. Wherever the first 2 intersect, that’s also where the third line will intersect the first 2.)

Engagement: Develop Effort and Persistence. Provide background knowledge to help focus on increasing the length of on-task orientation in the face of distractions. For example, provide images and definitions of obtuse, acute, and right triangles.
Supports accessibility for: Attention; Social-emotional skills

Student Facing

Each student in your group should choose 1 triangle. It’s okay for 2 students to choose the same triangle as long as all 3 are chosen by at least 1 student.

1. Construct the circumscribed circle of your triangle.
2. After you finish, compare your results. What do you notice about the location of the circumcenter in each triangle?

Anticipated Misconceptions

If students struggle to get started, ask them to look back at their work from the previous activity.

Activity Synthesis

The purpose of the discussion is to make informal observations about the locations of the circumcenters. Display this applet for all to see. Don’t immediately move the slider that controls the measure of one of the triangle’s angles.

Alternatively, display these images for all to see.

Ask students why it makes sense that the right triangle has its circumcenter on one of the triangle’s sides (we know that a right angle is inscribed in a half circle, so when you draw the hypotenuse of the triangle, it’s the diameter of the circle).

If using the applet, move the slider to show how the position of the circumcenter changes as the angle moves between being obtuse and acute. Ask students to observe and explain what happens to the circumcenter. (When the angle measure is greater than 90 degrees, the arc in which the angle is inscribed grows to more than 180 degrees. The triangle therefore takes up less than half the circle, so the circumcenter must be outside the circle. Alternatively, when the angle gets smaller, the circle “shrinks” to fit around the triangle. The arc in which the angle is inscribed measures less than 180 degrees, so the triangle fits more snugly in the circle and the circumcenter is inside the triangle.)

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Give groups additional time to make sure that each student can explain what they noticed about the location of the circumcenter in each triangle. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of observations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

The goal is to reinforce these 3 concepts: All triangles have circumscribed circles, the center of the circle is the intersection of the perpendicular bisectors of the triangle’s sides, and the circumcenter is equidistant from the triangle’s vertices.

Display this image for all to see.

Tell students that the points represent locations of 3 research stations in a desert, and that scientists want to build a supply hut that is the same distance from all 3 stations. What should they do? Challenge students to be precise in their use of language, and to explain why their ideas will work. Be sure these points come up in the discussion:

• The scientists should create the perpendicular bisectors of the sides of triangle $$RST$$.
• The perpendicular bisectors will meet in one point.
• This point is equidistant from all the vertices of the triangle, because points on perpendicular bisectors of a segment are equidistant from the segment’s endpoints.

Next, tell students that the scientists want to build another research station the same distance from the supply hut as the other stations. What should they do? Be sure these points come up in the discussion:

• The scientists should construct the circumscribed circle and put the new station anywhere on the circle.
• We know this triangle has a circumscribed circle because all triangles have one.
• If the station goes on the circle, it will be the same distance from the supply hut as the others because all points on a circle are the same distance from the circle’s center.

5.4: Cool-down - Fair Placement (5 minutes)

Cool-Down

Suppose we have triangle $$ABC$$ and we construct the perpendicular bisectors of all 3 sides. These perpendicular bisectors will all meet at a single point called the circumcenter of the triangle (label it $$D$$). This point is on the perpendicular bisector of $$AB$$, so it’s equidistant from $$A$$ and $$B$$. It’s also on the perpendicular bisector of $$BC$$, so it’s equidistant from $$B$$ and $$C$$. So, it is actually the same distance from $$A,B,$$ and $$C$$. We can draw a circle centered at $$D$$ with radius $$AD$$. The circle will pass through $$B$$ and $$C$$ too because the distances $$BD$$ and $$CD$$ are the same as the radius of the circle.
In this case, the circumcenter ​​​happened to fall inside triangle $$ABC$$, but that does not need to happen. The images show cases where the circumcenter is inside a triangle, outside a triangle, and on one of the sides of a triangle.