Lesson 3

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.

Scientific calculators are required if students will do the extension for the activity A Particular Perpendicular.