# Lesson 6

A Special Point

## 6.1: Notice and Wonder: Salt Pile (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the question, “Why does the salt pile up to make ridges and a peak?” which will be useful when students study triangle incenters in an upcoming activity. While students may notice and wonder many things about these images, the peak and ridges formed are the important discussion points.

This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

### Launch

If desired, demonstrate the process of pouring salt on a triangle. To do so, cut a triangle out of cardboard. Place a cup or bottle on top of a plate, and set the triangle on top. Pour salt on the triangle slowly and keep pouring after the triangle has reached capacity to show how the salt falls.

Alternatively, show students this video.

Then, display the images in the task statement 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 with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

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 concept of the distance to the sides of the triangles as a factor in the direction a given grain of salt will fall does not come up, ask students to discuss this idea.

## 6.2: Point and Angle (15 minutes)

### Activity

In this activity, students show that a point is on an angle bisector if and only if it is equidistant from the rays that form the angle. This concept is essential for the next activity, where students reason that the 3 angle bisectors of a triangle meet at a single point that is equidistant from each side of the triangle.

### Launch

The questions here are designed to help students visualize the distance from a point to the 2 sides of an angle, and to develop the conjecture that an angle bisector is the set of points equidistant from the sides of an angle. Display the applet for all to see.

Ask students whether it is possible to fit the circles between the two rays so that the rays are tangent to the circles, then move the circles inside the rays to demonstrate.

Click the button labeled “centers” and ask students what they notice. (They may notice that the centers of the circles appear to be collinear with each other and point $$A$$.)

Click the button labeled “radii” and ask students what they notice. (They may notice that the rays are equidistant from the circle centers, and they may notice or recall that these rays must be tangent to the circles or perpendicular to the radii.)

Ask students why the radii are drawn at an angle to each other, instead of forming a straight line. (To measure the distance from a point to the angle’s sides, we need to draw segments perpendicular to the lines.)

Finally, ask students, “The centers of the circles appear to lie on a line. Suppose we drew a line passing through all the centers. How does it seem like it might relate to the angle?” (It seems like it would bisect the angle.)

Engagement: Develop Effort and Persistence. Break the class into small discussion groups. Invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention

### Student Facing

Here is an angle $$BAC$$ with 2 different sets of markings.

1. Point $$E$$ is the same distance away from each side of angle $$BAC$$. Make a conjecture about angles $$EAB$$ and $$EAC$$ and prove it.
2. Point $$H$$ is on the angle bisector of angle $$BAC$$. What can you prove about the distances from $$H$$ to each ray?

### Launch

Display this image for all to see.

Ask students these questions, designed to help them visualize the distance from a point to the 2 rays of an angle:

• “Is point $$F$$ closer to ray $$AC$$ or ray $$AB$$?” (It is closer to ray $$AC$$.)
• “Is point $$E$$ closer to ray $$AC$$ or ray $$AB$$?” (It is hard to tell. It looks like it might be the same distance from both rays.)
• “How could we verify that point $$E$$ is the same distance from the 2 rays of the angle?” (We could measure the distance from point $$E$$ to the rays by drawing segments passing through point $$E$$ perpendicular to the rays.)
Engagement: Develop Effort and Persistence. Break the class into small discussion groups. Invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention

### Student Facing

Here is an angle $$BAC$$ with 2 different sets of markings.

1. Point $$E$$ is the same distance away from each of the 2 rays that form angle $$BAC$$. Make a conjecture about angles $$EAB$$ and $$EAC$$ and prove it.
2. Point $$H$$ is on the angle bisector of angle $$BAC$$. What can you prove about the distances from $$H$$ to each ray?

### Anticipated Misconceptions

Students may need to be reminded that an angle bisector divides an angle into 2 congruent halves.

### Activity Synthesis

The key point for discussion is that all points equidistant to the two rays are on the angle bisector, and all points on the angle bisector are equidistant to the two rays. Here are some questions for discussion:

• “How does this relate to the salt pile activity?” (If the angle were one of the angles in the triangle in the salt pile, the angle bisector would represent the ridge. The salt that forms the ridge is the same distance from either side, so it doesn’t fall in one direction or the other.)
• “What is the difference between what you showed in the first question and what you showed in the second question?” (In the first question, we showed that if a point is equidistant from the rays that form an angle, then it’s on the angle bisector. In the second, we proved the converse: If a point is on the angle bisector, it’s equidistant from the rays that form the angle.)
• “Have we proven these conjectures for all angles or just this one?” (This works for all angles, because we didn’t rely on any specific measurements or placements. If we drew a new angle, the same arguments would all apply.)

Tell students that what they’re learning will be useful when they construct another special circle in an upcoming lesson.

## 6.3: What If There Are Three Sides? (15 minutes)

### Activity

In a previous lesson, students reasoned that the perpendicular bisectors of a triangle’s sides meet at a single point. Here, students use a similar line of argument, combined with the results from the previous activity about points on angle bisectors, to determine that the 3 angle bisectors of a triangle also meet at a single point. This point is called the triangle’s incenter. The fact that this point is the same distance from all 3 sides of a triangle will lead to the construction of an inscribed circle in the next lesson.

### Launch

Remind students that they have shown that all the perpendicular bisectors of a triangle met at a single point, and that point was the same distance from all the vertices of the triangle. Tell them they’re going to try and see if something similar is true for the angle bisectors of a triangle.

Give students 2–3 minutes to work. Then pull them back together to make sure all students have correctly sketched segments showing the distance between point $$G$$ and the sides of the triangle.

### Student Facing

Two angle bisectors have been constructed in triangle $$ABC$$. They intersect at point $$G$$.

1. Sketch segments that show the distances from point $$G$$ to each side of the triangle.
2. How do the distances from point $$G$$ to sides $$AB$$ and $$BC$$ compare? Explain your reasoning.
3. How do the distances from point $$G$$ to sides $$AC$$ and $$BC$$ compare? Explain your reasoning.
4. Will the third angle bisector pass through point $$G$$? Explain your reasoning.

### Student Facing

#### Are you ready for more?

What shape would the ridge form if you poured salt onto a piece of cardboard with a long straight edge and a small hole cut out of it? Explain your reasoning.

### Anticipated Misconceptions

If students are having trouble sketching segments that show the distance from point $$G$$ to the sides of the triangle, suggest that they use an index card to estimate a right angle.

### Activity Synthesis

Tell students that this point where the angle bisectors meet is called the triangle’s incenter. We will add a theorem about a triangle’s incenter to the reference chart in the next lesson, after we look at a special circle related to incenters.

The goal is to further explore how to visualize the distance between a point and the side of a triangle, in order to strengthen students’ understanding that the incenter of a triangle is the same distance from all 3 of the triangle’s sides. First, ask students, “How does the incenter relate to salt pile?” (This point is the same distance from all the sides of a triangle, so grains of salt that land on this point balance there and don’t fall towards any of the sides.)

Then, display this image for all to see, explaining that the dashed lines are the angle bisectors for triangle $$ABC$$.

For each point $$D,E,F,$$ and $$G,$$ ask students these questions: Which side or sides are closest to the point? Which sides are equidistant from the point, if any? Sample responses:

• Point $$D$$ is closest to side $$AC$$. None of the distances to the other sides are equal.
• Point $$E$$ is closest to side $$BC$$. It is equidistant from sides $$AB$$ and $$AC$$.
• Point $$F$$ is equidistant from sides $$AC$$ and $$BC$$. It is closer to these 2 sides than it is to side $$AB$$.
• Point $$G$$ is equidistant from all 3 sides.
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares their reasoning for the sides that are closest to or equidistant from each point, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” Call students’ attention to any words or phrases that helped to clarify the original statement. For example, “Point $$E$$ is equidistant from sides $$AB$$ and $$AC$$ because it lies on the angle bisector of angle $$BAC$$.”
Design Principle(s): Support sense-making
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of incenters.
Supports accessibility for: Conceptual processing; Language

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students proved that an angle bisector is the set of points equidistant from the rays that form the angle, and used that concept to find the incenter of a triangle. Display an image of a segment and its perpendicular bisector alongside an image of an angle with its angle bisector:

Invite students to compare and contrast angle bisectors and perpendicular bisectors. (Each cuts something in half. The perpendicular bisector cuts a segment in half, while an angle bisector cuts an angle in half. Both structures have to do with points being the same distance away from 2 objects. The perpendicular bisector is the set of points equidistant from the endpoints of a segment, whereas the angle bisector is the set of points equidistant from the rays that form the angle. Both divide a region into sets of points closer to one object than another object.)

## Student Lesson Summary

### Student Facing

Salt piles up in an interesting way when poured onto a triangle. Why does that happen?

As the salt piles up and reaches a maximum height, new grains of salt will fall off toward whichever side of the triangle is closest. We can show that points on an angle bisector are equidistant from the rays that form the angle. So, salt grains that land on an angle bisector will balance and not fall towards either side. This is why we see ridges form in the salt.

In the images, segments $$RT,AD,BD,$$ and $$CD$$ are angle bisectors. This means that point $$T$$ is the same distance from ray $$RQ$$ as it is from ray $$RS$$. In triangle $$ABC$$, point $$D$$ is the same distance from all 3 sides of the triangle—it’s the triangle’s incenter.