In this Warm-up, students consider an image with a line and a circle on the same coordinate plane. They consider mathematical questions that could be asked about the image, which will be useful in a following activity when they find the points of intersection of circles and lines.

Invite several partners to share one question with the class, and record their responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goals, such as “point of intersection,” “perpendicular,” or “parallel.”

In this activity, students graph a circle and line, and then use the graph to identify points of intersection. They also verify these intersection points using algebraic methods. This is an example of solving a system consisting of a linear equation and a quadratic equation in two variables graphically, although students are not expected to use that language to describe their process in this lesson. 

Monitor for students who use different strategies to verify that the intersection points lie on both the line and the circle. Some approaches students may use include:

• Substitute the points directly into the circle equation, and then verify that the equation is true
• Use the Pythagorean Theorem to determine that the distance from the point to the center of the circle is the same as the radius of the circle.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students apply their understanding of slopes of parallel and perpendicular lines to problems involving lines and circles. They also consider possible intersections of circles and lines. As students create their own equations that meet particular conditions, they are making use of the structure of the equations of parallel and perpendicular lines and of equations of circles (MP7).

Monitor for students using different strategies to identify the second intersection point of the line and circle. Some approaches students may use include:

• Graphing the line and circle and identifying the point of intersection on the graph.
• Verifying that the line has a -intercept at or rewriting the equation in slope-intercept form to see the intercept.
• Using a slope triangle to identify points on the line until they find one that is also on the circle.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).