Lesson 4

In this activity, students practice subtracting coordinates as part of a method for calculating the distance between two points. This will be helpful in an upcoming activity in which students build the equation of a circle by generalizing the process for finding distance on a coordinate grid.

Monitor for students who subtract pairs of coordinates directly without first finding the coordinates of point \(N\).

Arrange students in groups of 2. Provide access to scientific calculators. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

If students are stuck, suggest that they find the coordinates of point \(N\).

Invite a previously identified student who subtracted coordinates directly to share their process for finding the lengths of the legs. Emphasize the use of subtraction to find the lengths of the legs. Consider displaying the image from the task and labeling the legs \(22-12=10\) and \(53-8=45\).

Students use the Pythagorean Theorem to test whether points are on a circle with a given center and radius. They will build on this process in the next activity to develop a general equation for a circle.

If students have trouble getting started, remind them of the definition of a circle: It is the set of points the same distance away from a given center.

Display this image for all to see.

Draw a right triangle on the image that could be used to find the distance between the center of the circle and the point shown. Ask students how they would find the lengths of the legs. As students describe the process, label the lengths of the legs \(x-6\) and \(y-10\).

Students build the general equation of a circle. As students generalize the work of the previous activities, they are making sense of a problem (MP1).

Monitor for students who write \((\text{-}3 - x)^2\) instead of \((x - (\text{-}3))^2\) in the first equation. Explain that either one is correct, but we traditionally write the variable as the first term in the binomial.

Tell students that we traditionally use the letter \(r\) to stand for the radius of a circle, and we use the letters \((h,k)\) to stand for the center of a circle.

If students struggle to write the first equation, suggest that they draw a right triangle that could be used to find the distance between \((x,y)\) and the center of the circle.

Display this image for all to see.

Ask students to connect their general equation to the circle. As students give their descriptions, plot a point \((x, y)\) and draw a right triangle that can be used to find the distance between \((x,y)\) and \((h,k)\). Label the legs \((x-h)\) and \((y-k)\), and label the hypotenuse \(r\).

Tell students that just like an equation in the form \(y=mx+b\) defines a line, an equation in the form \((x-h)^2+(y-k)^2=r^2\) defines a circle. The equation can be thought of as a point tester—any point \((x,y)\) that makes the equation true is a point on the circle, and conversely, all points on the circle make the equation true.