A common exercise that’s used to reinforce the concept that the tangent of a circle is perpendicular to its radius involves finding the radius of a circle given two measurements which are related to the tangent and the diameter secant. For example, students might be asked to find the radius of this circle:
In the treatments of this I’ve seen (including my current textbook), students are instructed to use the Pythagorean Theorem to solve. Specifically, in the example case, \[r^2 + 4^2 = (r + 2)^2 = r^2 + 4r + 4 \\ \Rightarrow 4^2 = 4r + 4 \Rightarrow r + 1 = 4 \Rightarrow \\ r = 3\]
I think, for typical students, this is the correct approach to start with. But students are then given a batch of these problems to solve, using the same method. This, I think, is a waste of both effort and opportunity.
Working with a student in tutoring yesterday, I solved the equation for r, since all the problems in this unit involve finding r. Letting t stand for the tangent segment and s for the external portion of the secant: \[r^2 + t^2 = (r + s)^2 = r^2 + 2rs + s^2 \\ \Rightarrow t^2 = 2rs + s^2 \Rightarrow 2rs = t^2 – s^2 \Rightarrow \\ r = \frac{t^2 – s^2}{2s} \]
Once students develop this formula, they can much more efficiently solve the same problems. In the example image, \(r = \frac{4^2 – 2^2}{2\times 2} = \frac{12}{4} = 3\).
The key, I believe, is making sure that students understand the relationship between this formula and the Pythagorean Theorem. It’s common practice, unfortunately, to teach the distance formula (\(D = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}\)) significantly before the Pythagorean Theorem, and to weakly tie them together, if at all.
Making the Pythagorean Theorem the axis point for measurements is crucial, but at the same time, reinforcing its flexibility by creating efficient forms of it also reinforces that math should be a malleable process.
I think once place we’ve stumbled in mathematics education is in putting the cart before the horse: We emphasize learning Yet Another Formula instead of learning base relationships and how to make those relationships more efficient for specific purposes.