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• “In the second question, what would happen if point were somewhere else on the angle bisector? What would change? What would stay the same?” (Again, the sizes of the right triangles are di erent, but the proof that they are congruent is exactly the same, which means the conclusion that point is equidistant to each ray will remain true.)
8.3 What If There Are Three Sides?
15 minutes
In the previous activity, students found that points on the angle bisector are equidistant to the two rays that form the angle. In this activity, a third side is introduced to form a triangle. Students nd points on an angle bisector that are closer to the two sides adjacent to the bisected angle and points that are closer to the opposite side. Between these extremes is a point that is the same distance to all three sides. This special point is called the incenter The fact that it is the same distance from all three sides of a triangle leads to the construction of the incircle in the next lesson.
Launch
Give students 5 minutes to work on question 1 before pausing for a brief, whole class discussion. Display the image from question 1 for all to see. Ask students what they know about points on segment (they are the same distance from sides and ). In one color, ask students to highlight points on that are closer to sides and than they are to side . In another color, ask students to highlight points on that are closer to side . Ask students to approximate a point on between the other points that is the same distance away from sides
, and . Ask students about the relationship between this point and angle (the point is on the angle bisector of angle because it is the same distance from sides and ). Give students 5 minutes to complete the task, followed by another whole-class discussion.
Student Task Statement
Here is a triangle .
1. Angle is bisected by segment .
,
Unit 7 Lesson 8: A Special Point
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