Algebra II Lesson

Circles, Tangents & Secants: From Graphs to Algebraic Systems

Shared by A STEM Educator

11th GradeThin-Slicing90 min

Learning Objective

Students will be able to identify tangent and secant lines relative to a circle on a coordinate plane, determine their equations from graphical information, and solve the system formed by a circle equation and a linear equation algebraically to find intersection points.

Lesson Overview

Students progress from visually identifying intersection points of tangents and secants on a graph to writing the equations of a circle and secant line and solving the resulting system algebraically. Tasks are anchored in a consistent real-world context of a skatepark design where curved ramps (circles) meet straight paths (tangents and secants).

Materials

Vertical whiteboards or chart paper
Dry-erase markers (multiple colors)
Random grouping tool
Printed or projected task cards
Graphing calculators or Desmos access (for verification only)
Rulers

Scaffolded Task Progression

1Task 1

The skatepark designer changed the skater’s route again. The bowl is still the same circle, but now the skater rides along a simpler slanted line. Use the graph to find the approximate coordinates of both points where the path intersects the bowl.

[ Interactive graph / diagram — available in the full lesson ]

2Task 2

Now the skater's path is different. Look at the two graphs below. For each one, state:
(A) how many points does the path meet the bowl, and(B) do you think the path is a TANGENT or a SECANT? (A tangent touches at exactly one point; a secant crosses at two points.) Graph 1: [ Interactive graph / diagram — available in the full lesson ] Graph 2: [ Interactive graph / diagram — available in the full lesson ]

3Task 3

The skatepark map now shows a non-horizontal path. Using the graph, find the coordinates of the intersection point(s) of the line and the circle. Then state whether the line is a tangent or a secant.

[ Interactive graph / diagram — available in the full lesson ]

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