In this lesson, students construct a line perpendicular to the radius of a circle that goes through the point where the radius intersects the circle. They prove that this line intersects the circle in exactly 1 point. That is, the line is tangent to the circle. Students also prove the converse: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
Students use these findings to show that an angle circumscribed about a circle is supplementary to the central angle defined by the points where the angle is tangent to the circle. As students write an explanation of this property, they are reasoning abstractly and quantitatively (MP2).
The concept of tangent lines will be useful in subsequent lessons when students construct circles inscribed in triangles. In future courses, tangent lines appear when calculating instantaneous rates of change and when defining periodic functions.
A particular choice of construction tools here and throughout this unit is not necessary. Paper folding and straightedge and compass moves are both acceptable methods.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
- Use the relationship between tangent lines and radii to prove (using words and other representations) a theorem about circumscribed angles.
- Let’s explore lines that intersect a circle in exactly 1 point.
Scientific calculators are required if students will do the extension for the activity A Particular Perpendicular.
- I can use the relationship between tangent lines and radii to calculate angle measures and prove geometric theorems.
- I know that a line tangent to a circle is perpendicular to the radius drawn to the point of tangency.
The part of a circle lying between two points on the circle.
An angle formed by two rays whose endpoints are the center of a circle.
A chord of a circle is a line segment both of whose endpoints are on the circle.
An angle formed by two chords in a circle that share an endpoint.
A line is tangent to a circle if the line intersects the circle at exactly one point.