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Lesson 7: Tangent Lines
• Understand that radii are perpendicular to lines tangent to a circle. Lesson Narrative
In this lesson, students prove that the shortest path between a point and a line is the segment connecting them that is perpendicular to the given line, and de ne the distance between the point and the line as the length of this segment. Students will then investigate the relationship between a line that intersects a circle at only one point without crossing, called a tangent line, and the radius of the circle going through that point. Students prove that lines tangent to circles are perpendicular to the radius at the point of tangency. These ideas will lead to the topic of incenters and incircles of triangles in the next two lessons.
Students create viable arguments and critique the reasoning of others when discussing proofs of various claims about tangent lines and distances (MP3).
Student Learning Goals
• Let’s explore lines tangent to circles.
7.1 Distance From a Point to a Line
Warm Up: 5 minutes
The purpose of this warm-up is to use physical intuition to develop the idea that the shortest path between a point and a line is the path perpendicular to the line going through the point. Students will prove this fact in the next activity and use this fact to prove that radii are perpendicular to tangents in a later activity.
Students attend to precision when they explain what it means to ask about the distance between a point and a line (MP6).
Student Task Statement
Line represents a straight part of the shoreline at the beach. If you were in the ocean at point and you wanted to get to the shore as fast as possible, what path would you take? Explain your reasoning.
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Teacher Guide