Lesson 3

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).

Arrange students in groups of 2. Give students 1 minute of quiet work time and then time to share their work with a partner.

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 \(C\) to the shore. So, there are many possible distances between point \(C\) and line \(\ell\). 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 \(C\) and line \(\ell\).)
• “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.)

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.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

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 in exactly 1 point. Display this image for all to see.

Ask students:

• “Name any tangent lines in this diagram.” (Line \(RS\) is a tangent line.)
• “Why isn’t line \(PQ\) a tangent line?” (It intersects the circle in 2 points.)
• “Why isn’t segment \(ON\) a tangent line?” (The segment intersects the circle in 1 point, but if it were extended out into a line it would intersect the circle twice.)

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 in more than 1 point, ask them to imagine that it does intersect the circle in 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. Here, they will prove the converse. Display this image for all to see:

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

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

Tell students that we call the point where the tangent line intersects the circle the point of tangency. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

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.

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 1 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 \(a+(b+d)+(c+e)+f\). But we also know that \(a+b+c\) and \(d+e+f\) are each equal to 180. So we can rewrite the first expression as \(a+b+c+d+e+f\), which is equal to \(180 + 180\) or 360 degrees.)

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 into the quadrilateral they have created.

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: Use their compass to draw a circle. Sketch an angle whose sides are approximately tangent to the circle, and label its measure \(a\). Draw radii to the points of tangency and label the central angle’s measure \(b\). Then, ask students:

• “How does the picture change as the value of \(a\) changes?” (For small values of \(a\), the angle is “far away” from the circle, and the central angle is almost a straight line. For large values of \(a\), the angle is very “close” to the circle.)
• “What is the largest possible value for \(a\)?” (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 \(a=179\)?” (The angle would almost be a tangent line. The central angle would be a tiny, 1 degree angle.)