Unit 7, Lesson 7
Tangent Lines
Integrated Math 2
7.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this Warm-up is to revisit 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 use this fact to prove that radii are perpendicular to tangents in a subsequent activity.
Students attend to precision when they explain what it means to ask about the distance between a point and a line (MP6).
Launch
Arrange students in groups of 2. Give students 1 minute of quiet work time and then time to share their work with a partner.
Activity
NoneLine represents a straight part of the shoreline at a beach. Suppose you are in the ocean at point and you want to get to the shore as fast as possible. Assume there is no current. Segments and represent two possible paths.
Diego says, “No matter where we put point , the Pythagorean Theorem tells us that segment is shorter than segment . So segment represents the shortest path to shore.”
Do you agree with Diego? Explain your reasoning.
Student Response
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Building on Student Thinking
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. Display this image for all to see.
Ask students:
- “There are many possible paths from your location at point to the shore. So there are many possible distances between point and line . If someone asks how far away you are from the shore, what do they mean?” (They are asking for the shortest possible distance between point and line .)
- “What does Diego’s reasoning tell us about the shortest path from a point to a line?” (It is measured along a line through the point perpendicular to the line.)
7.2
Activity
Instructional Routines
NoneMaterials
Activity Narrative
Students construct a line perpendicular to the circle’s radius that passes through the point where the radius intersects the circle. They prove that this line is a tangent line. In the Activity Synthesis, the class proves that the converse is also true. As students construct lines with given properties on the circle, they are making sense of the problems involving tangent lines (MP1).
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
First, tell students that here and in other subsequent activities, paper folding, compass and straightedge, and digital geometry tools are all valid construction methods.
Then tell students that a line tangent to a circle intersects the circle at exactly 1 point. Display this image for all to see.
Ask students:
- “Name any tangent lines in this diagram.” (Line is a tangent line.)
- “Why isn’t line a tangent line?” (It intersects the circle at 2 points.)
- “Why isn’t segment a tangent line?” (The segment intersects the circle at 1 point, but if it were extended out into a line, it would intersect the circle at 2 points.)
7.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
Students use their new understanding of tangent lines to prove a property of circumscribed angles. It is not necessary to use the word “circumscribed” at this time. This term will be used in the context of circumscribed circles in a subsequent activity. Students use the properties they discovered in an earlier activity as they persevere in solving problems involving tangent lines (MP1).
Launch
Give students a few minutes of quiet work time. Then, if students are struggling with the sum of the measures of the angles of a quadrilateral, draw a quadrilateral that isn’t regular for all to see. Draw one of the quadrilateral’s diagonals. Label the angles of the resulting triangles as they’re labeled in the image.
Ask students how this drawing can help us figure out the sum of the measures of the angles in a quadrilateral. (The sum of the measures of the quadrilateral’s angles can be written . But we also know that and are each equal to 180. So we can rewrite the expression as , which is equal to , or 360 degrees.)
Lesson Synthesis
Display this image for all to see. Tell students that line is tangent to the circle at point .
Ask students to find as many right angles as they can in the diagram and to explain their reasoning. Here are some points for discussion:
- Angles and are each right angles. We know that line is perpendicular to radius because it is tangent to the circle.
- Angles and are each right angles. Each is an inscribed angle whose associated central angle is a diameter. These central angles measure 180 degrees, so the inscribed angles must measure half that, or 90 degrees.
If students are wondering how the idea of a “tangent line” relates to the “tangent” they learned about in trigonometry, tell them that, in future courses, they will extend trigonometry beyond right triangles and at that point the two notions of “tangent” will connect.
Student Lesson Summary
A line is said to be tangent to a circle if it intersects the circle at exactly 1 point. Suppose line is tangent to a circle centered at . Draw a radius from the center of the circle to the point of tangency, or the point where line intersects the circle. Call this point . It looks like radius is perpendicular to line . Can we prove it?
Every other point on the tangent line is outside the circle, so they must all be further away from the center than the point where the tangent line intersects the circle. This means the point where the tangent line intersects the circle is the closest point on the line to the center point . The radius must be perpendicular to the tangent line because the shortest distance from a point to a line is always along a perpendicular path.