3.1: Swim to Shore (5 minutes)
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.
Line \(\ell\) represents a straight part of the shoreline at a beach. Suppose you are in the ocean at point \(C\) and you want to get to the shore as fast as possible. Assume there is no current. Segments \(CJ\) and \(CD\) represent 2 possible paths.
Diego says, “No matter where we put point \(D\), the Pythagorean Theorem tells us that segment \(CJ\) is shorter than segment \(CD\). So, segment \(CJ\) represents the shortest path to shore.”
Do you agree with Diego? Explain your reasoning.
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.
- “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.)
3.2: A Particular Perpendicular (15 minutes)
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.
- “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.)
- Draw a radius in the circle. Mark the point where the radius intersects the circle and label it \(A\).
- Construct a line perpendicular to the radius that goes through point \(A\). Label this line \(n\).
- Line \(n\) intersects the circle in exactly 1 point, \(A\). Why is it impossible for line \(n\) to intersect the circle in more than 1 point?
- What kind of line, then, is \(n\)?
Are you ready for more?
Here is a circle centered at \(O\) with radius 1 unit. Line \(AB\) is tangent to the circle.
- Calculate the length of segment \(AB\).
- How does the length of segment \(AB\) relate to the tangent of 50 degrees? Why is this true?
- In right triangle trigonometry, “tangent” is defined in terms of side ratios. Write another definition of tangent (in the sense of a numerical value, not a line) in terms of a circle with radius 1 unit.
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)
Supports accessibility for: Conceptual processing; Visual-spatial processing
3.3: Another Angle (15 minutes)
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.)
The image shows an angle whose rays are tangent to a circle.
- Mark the approximate points of tangency.
- Draw the 2 radii that intersect these points of tangency. Label the measure of the central angle that is formed \(w\).
- What is the value of \(w+z\)? Explain or show your reasoning.
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.)
Design Principle(s): Support sense-making; Optimize output (for justification)
Display this image for all to see. Tell students that line \(CD\) is tangent to the circle at point \(B\).
Ask students to find as many right angles as they can in the diagram and to explain their reasoning. Sample responses:
- Angles \(ABC\) and \(ABD\) are each right angles. We know that line \(CD\) is perpendicular to radius \(AB\) because it is tangent to the circle.
- Angles \(BGF\) and \(BHF\) 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 2 notions of “tangent” will connect.
3.4: Cool-down - Tangents and Triangles (5 minutes)
Student Lesson Summary
A line is said to be tangent to a circle if it intersects the circle in exactly 1 point. Suppose line \(\ell\) is tangent to a circle centered at \(A\). Draw a radius from the center of the circle to the point of tangency, or the point where line \(\ell\) intersects the circle. Call this point \(B\). It looks like radius \(AB\) is perpendicular to line \(\ell\). 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 intersects. This means the point \(B\) where the tangent line intersects the circle is the closest point on the line to the center point \(A\). The radius \(AB\) must be perpendicular to the tangent line because the shortest distance from a point to a line is always along a perpendicular path.