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).

If students struggle to begin to construct the perpendicular line, suggest they extend the radius beyond the circle.

If students get stuck on explaining why it is impossible for their line to intersect the circle at more than 1 point, ask them to imagine that it does intersect the circle at more than 1 point. Then part of the line would be inside the circle. What do students know about distances and perpendicular lines that makes this impossible?

In the activity, students showed that a line perpendicular to a radius through the point where the radius intersects the circle is a tangent line. The goal of this discussion is for students to prove the converse. Display this image for all to see:

Tell students that line is tangent to the circle. Draw radius . Tell students that we want to prove that radius is perpendicular to line . Ask, “How can distances help us prove this?” (If we show that is the shortest path from point to line , then and are perpendicular.)

Invite students to imagine a point on line that is closer to point than  is. Ask, “Why is this impossible?” (If is closer than to line , then must be inside the circle. Therefore, line must pass through the circle. In that case line  would intersect the circle twice. But by definition, a tangent line touches the circle only once. Radius  therefore measures the shortest distance between point and line .)

Tell students that we call the point where the tangent line intersects the circle the “point of tangency.” Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency. (Theorem)

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). 

If students struggle to begin, ask them to look back to the previous activity to determine if there are any angle measures they can mark on the quadrilateral they have created.

The goal of this discussion is for students to understand that the circumscribed angle and central angle formed by tangent lines and radii of a circle are supplementary.

Ask several students to describe their reasoning. If time permits, invite students to create an image similar to the one in the activity, but with larger or smaller angles: Display these instructions for students to follow: 

1. Use a compass to draw a circle. 
2. Sketch an angle whose sides are approximately tangent to the circle, and label its measure .
3. Draw radii to the points of tangency, and label the central angle’s measure .

Then, ask students:

• “How does the picture change as the value of a changes?” (For small values of , the angle is “far away” from the circle, and the central angle is almost a straight line. For large values of , the angle is very “close” to the circle.)
• “What is the largest possible value for ?” (There is no “largest” value. The angle needs to measure less than 180 degrees. Examples of very large circumscribed angles would be 179.9 degrees or 179.99 degrees.)
• “What would the picture look like for ?” (The angle would almost be a tangent line. The central angle would be a tiny, 1-degree angle.)