Tricky Tangent and Secant Lines

After completing (ex 3), which asks students to apply their knowledge of the ratio relating intersecting chords and lines in circles, teachers can introduce the second main idea of the lesson:  a tangent line and a secant line intersecting in a circle.  This is a full lesson, and instead of asking students to derive the proof for this relationship, a youtube video of a mathematics teacher discussing the mathematical reasoning behind this idea is explained in 3-4 minutes.  Teachers can then review how to complete examples 4 and 5 with students to highlight how these ratios are related.  I really like to emphasis that students must memorize “tangent-squared = outside(whole).”  Further, this is a great formula to write our when students start to complete these problems.

Lastly, teachers can introduce the last major concept for class, that two secants intersecting at a point outside a circle will create another ratio.  I really like to emphasis that students must memorize “outside(whole) = outside(whole).”  Teachers can then work through (ex 6) and (ex 7) with students.  The Do Now questions specifically review factoring because students will be required to factor in order to complete (ex 7).  We will also find similar examples with factoring in the practice examples. 

Students can work on practice questions, examples 1-12 in class notes for the rest of class.  Please note that questions #10 and #12 both require factoring and may be a great example to choose to review with students.  Before class is over, students can write and show their work of at least one or two examples on the board.  I would suggest that teacher specifically review #8, 9, and 11 before asking students to start their exit ticket.   Each of these questions review one of the three cases learned in class, and #10 or #12 can be reviewed as a reinforcement of factoring if time remains.

The exit ticket asks students to