Page 142 - IM_FL_Geometry_Print Sample

Activity Synthesis
The purpose of discussion is to establish the fact that the shortest distance between a point and a line is the length of the perpendicular path between them. Ask students to share their responses to question 3. Push students to justify why the hypotenuse of a right triangle is longer than the legs. Let be the length of segment , be the length of segment , and be the length of segment
. Consider asking:
• “What do we know about lengths in this right triangle?” (We know by the Pythagorean Theorem.)
• “How do we know
• “It’s true that
?” (We know
?” (Since , so is minus a positive number.)
, but -5 is not greater than 3. How do we know that just because because and are both positive numbers and larger numbers
have larger squares.)
Ask students to add this theorem to their reference charts as you add it to the class reference chart:
The segment perpendicular to a line through a given point has the shortest length of any segment connecting the point and the line.
7.3 Radius, Meet Tangent
15 minutes
In this activity, students prove that a line tangent to a circle is perpendicular to the radius that meets the line at the point of tangency. This proof relies on the fact that the shortest path between a line and a point is perpendicular to the line. The de nition of tangent used in this activity is speci c only to the circle. It is beyond the scope of this course to rigorously de ne what it means for a line to be tangent to an arbitrary curve.
Instructional Routines
• Notice and wonder
What: This routine can appear as a warm-up or in the launch of a classroom activity. Students are shown some media or a mathematical representation. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to write down things they notice and things they wonder. After students have had a chance to write down their responses, the teacher asks several students to share things they noticed and things they wondered; these are recorded by the teacher for all to see. Usually, the teacher steers the conversation to wondering about something mathematical that the class is about to focus on.
Unit 7
Lesson 7: Tangent Lines 69