Students will learn how to find the tangent and normal line, given an equation and a point.

Equations of Tangents & Normals by Eddie Woo

Conclude by giving your students these challenges:

Solve \(x^2 + 19x - x! = 0.\)

Here's one solution. And here's another:

$$\begin{align} & x^2 + 19x - x! = 0 \\ & x^2 + 19x = x! \\ & x + 19 = (x - 1)! \\ & (x - 1) + 20 = (x - 1)! \\ & 1 + 20/(x - 1) = (x - 2)! \end{align}$$The right-hand side is an integer, so the left-hand side must also be an integer. Thus, \(x - 1\) must divide \(20.\) So \(x - 1\) must be \(1,\,2,\,4,\,5,\,10,\) or \(20.\) Thus, \(x\) must be \(2,\,3,\,5,\,6,\,11,\) Testing each value, we find \(x = 5\) is the only value that works.

Find the radius of the circle:

Prereqs include knowing a rectangle has four right angles, knowing the intersecting chords theorem, and knowing the Pythagorean theorem. Here's the solution. Before watching this video, you should be aware it contains one mistake. He uses Thales's theorem to conclude AE is a diameter, but Thales's theorem can only be used to prove an angle is \(90^\circ,\) subject to certain conditions. In this case, we know AE is a diameter because ABE is a right angle. If it's not obvious, you could imagine copying the path ABE, rotating it 180 deg around the center of the circle. What you get is a rectangle circumscribed by the circle. Thus AE is the diagonal of the rectangle, and O must be its midpoint, as any rectangle and its circumcircle have the same center, by symmetry.