Circles in Triangles
7.1: The Largest Circle (5 minutes)
Students start to explore the idea of a circle inscribed in a triangle.
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Use a compass to draw the largest circle you can find that fits inside each triangle.
Invite several students to share their results. Ask students, “Was one circle easier to work with than the other?” Students are likely to have found the obtuse triangle more challenging.
7.2: The Inner Circle (15 minutes)
In this activity, students create an arbitrary triangle and use what they know about angle bisectors and constructions to find the incenter. Then they construct segments measuring the distance from the incenter to the sides of the triangle. Finally, they construct the triangle’s inscribed circle, or the circle that is tangent to all 3 sides of the triangle.
Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).
If students have access to dynamic geometry software, suggest that it might be a helpful tool in this activity.
Supports accessibility for: Conceptual processing; Organization
- Mark 3 points and connect them with a straightedge to make a large triangle. The triangle should not be equilateral.
- Construct the incenter of the triangle.
- Construct the segments that show the distance from the incenter to the sides of the triangle.
- Construct a circle centered at the incenter using one of the segments you just constructed as a radius.
- Would it matter which of the three segments you use? Explain your thinking.
Remind students that the circle they constructed is said to be inscribed in the triangle. Students previously discussed inscribed circles in the Constructions and Rigid Transformations unit. Ask students to add this theorem to their reference charts as you add it to the class reference chart:
The 3 angle bisectors of a triangle meet at a single point, called the triangle’s incenter. This point is the center of the triangle’s inscribed circle. (Theorem)
Display several students’ inscribed circles for different kinds of triangles for all to see. The goal of the discussion is to draw conclusions about inscribed circles. Start by asking, “What do you notice about the triangles and their inscribed circles?” Sample responses:
- Some inscribed circles take up a large fraction of the area, while others take up a small fraction.
- The circle seems to be tangent to the sides of the triangles.
- The original triangles are each divided into 3 smaller triangles. For each small triangle, the base is a side of the original triangle and the height is the radius of the inscribed circle.
- The original triangles are each divided into 3 pairs of congruent right triangles with an angle that is half the measure of an angle in the original triangle.
Next, ask students:
- “How do we know that an inscribed circle is tangent to all 3 sides of the triangle?” (The radius that represents the distance from the incenter to each side of the triangle is by definition perpendicular to that side. A radius is perpendicular to a line at a point on the circle’s circumference only if the line is tangent to the circle at that point.)
- “Is it possible to draw a circle that is larger than the inscribed circle but still fits completely in the triangle?” (No. The circle has already expanded to fit tightly into the triangle. If the radius of the triangle were increased, the circle would push outside the triangle. The inscribed circle is the largest one that can fit inside the circle.)
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
7.3: Equilateral Centers (15 minutes)
Students prove that in an equilateral triangle, the incenter and the circumcenter coincide. Monitor for students who use triangle congruence and for those who use transformations.
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Supports accessibility for: Conceptual processing; Language; Visual-spatial processing
Design Principle(s): Optimize output (for justification); Cultivate conversation
The image shows an equilateral triangle \(ABC\). The angle bisectors are drawn. The incenter is plotted and labeled \(D\).
Prove that the incenter is also the circumcenter.
Are you ready for more?
- Suppose we have an equilateral triangle. Find the ratio of the area of the triangle’s circumscribed circle to the area of its inscribed circle.
- Is this ratio the same for all triangles? Explain or show your reasoning.
If students aren’t sure how to start, ask them what needs to be true about the segments \(DA,DB,\) and \(DC\) in order for point \(D\) to be the triangle’s circumcenter (the circumcenter is the single point equidistant from all the vertices, so if segments \(DA,DB,\) and \(DC\) are all congruent, then \(D\) is the circumcenter).
Then, ask students:
- Are there any auxiliary lines you can add to the diagram?
- Are there any transformations that might be helpful?
- Do any triangle congruence theorems apply?
- Are there any angle markings you can add to the diagram?
Invite students to share their reasoning. If possible, select a student who used triangle congruence and another who used transformations. Then, ask students:
- “Why doesn’t this proof work for a triangle that is not equilateral?”
- The angle measures in the original triangle wouldn’t be congruent, so the 6 small triangles created by drawing segments representing the distance from \(D\) to the triangle’s sides wouldn’t all be congruent.
- If \(AC\) and \(BC\) don’t have the same length, the perpendicular bisector of \(AB\) won’t pass through \(C\).
- “What will the inscribed and circumscribed circles look like for this triangle? Use your compass to construct them.” (They will be concentric circles.)
- “How would the perpendicular bisectors of triangle’s sides relate to the lines already drawn?” (They would overlap the segments showing the distance from point \(D\) to the sides of the triangle.)
The purpose of the discussion is to summarize characteristics of incenters and circumcenters. Ask students, “Compare and contrast the circumcenter and incenter of a triangle. When might you use one rather than another?”
- They are each centers of special circles.
- They both have to do with being an equal distance away from 3 objects.
- The circumcenter is the point that is the same distance away from the 3 vertices of a triangle, whereas the incenter is the point that is the same distance away from the 3 sides of the triangle.
- In an equilateral triangle, the incenter and the circumcenter are the same point.
- The circumcenter is the center of the triangle’s circumscribed circle. The incenter is the center of the triangle’s inscribed circle.
- The circumcenter is obtained by finding the intersection of perpendicular bisectors, whereas the incenter is obtained by finding the intersection of angle bisectors.
- Problems that require a point to be the same distance away from 3 other points will use a circumcenter, whereas problems that require a point to be the same distance away from 3 sides of a triangle will use an incenter.
7.4: Cool-down - Circular Table Top (5 minutes)
Student Lesson Summary
We have seen that the incenter of a triangle is the same distance from all 3 sides of the triangle. If we draw the congruent segments representing the distance from the incenter to the triangle’s sides, we can think of them as radii of a circle centered at the incenter. This circle is the triangle’s inscribed circle.
In this diagram, segments \(BD,CD,\) and \(AD\) are angle bisectors. Point \(D\) is the triangle’s incenter, and the circle is inscribed in the triangle.
The inscribed circle is the largest possible circle that can be drawn inside a triangle. Also, the 3 radii that represent the distances from the incenter to the sides of the triangle are by definition perpendicular to the sides of the triangle. This means the circle is tangent to all 3 sides of the triangle.