Page 168 - IM_FL_Geometry_Print Sample

Student Task Statement
Here is triangle with its incircle. Segments and are radii of the incircle that intersect the sides of the triangle.
Suppose the radian measure of angle , and the radian measure of angle
Write an equation that relates and your reasoning.
Student Response
is is .
. Explain
or equivalent. The sum of the angles in triangles and each add up to radians, so the sum of angles in quadrilateral must sum to . Angles and must both be right angles, though, because the radii are always perpendicular to tangents. This means together they add to . That leaves for the sum of the other two angles.
Activity Synthesis
Ask students to share their responses. The key result from this task is that the two angles in question are supplementary. Ask students what the equation radians means in terms of degrees. Use tracing paper to illustrate that the angles can be made to form a straight angle.
Lesson Synthesis
In this lesson, students constructed the incircle of an arbitrary triangle and applied what they know about circles to investigate circumscribed angles. Here are some questions for discussion:
• “Do all triangles have an incircle?” (Yes. It’s impossible for angle bisectors to be parallel, so there will always be a point that is equidistant to two di erent pairs of sides. If the rst pair of sides are sides and , and the second pair of sides are and , then the point in question is the same distance from sides and , and the same distance from sides and . That means the point must be an incenter that is the same distance from and and . This distance is the radius of the incircle.)
• “What would be the same or di erent if you tried to inscribe a circle inside a quadrilateral instead?” (In a quadrilateral, it is not guaranteed that all the angle bisectors meet at a point. It will always be possible to make a circle tangent to 3 of the 4 sides using the intersection of 2 angle bisectors. To give an example of a quadrilateral that can’t have a circle tangent to all 4 sides, look at a rectangle whose length is much much longer than its width. For a circle to be tangent to three of the sides, its radius has to be small compared to the width. But that means it won’t reach all the way across the rectangle to be tangent to the 4th side.)
Unit 7
Lesson 9: Circle Inside a Triangle 95