First, students will learn why a radius \(\overline{AB}\) is perpendicular to a segment \(\overline{CD},\) if and only if \(\overline{AB}\) bisects \(\overline{CD}.\) To begin, students will learn why the forward direction of the theorem is true. That is, if a radius \(\overline{AB}\) is perpendicular to a segment \(\overline{CD},\) then \(\overline{AB}\) bisects \(\overline{CD}.\)
Proof: perpendicular radius bisects chord by Khan Academy
Next, students will learn why the other direction of the theorem is true. That is, if a radius \(\overline{AB}\) bisects a segment \(\overline{CD},\) then \(\overline{AB}\) is perpendicular to \(\overline{CD}.\)
Proof: radius is perpendicular to a chord it bisects by Khan Academy
After that, students will learn how to use this theorem with the Pythagorean theorem, to find the distance from the center of a circle to a chord. Students will be given the radius or diameter of the circle, and the length of the chord, then asked for the distance from the center of the circle to the chord. For example: Find the distance from the center of a circle with a diameter of 26 in to a chord of 10 in (video). Also, students will learn how to find the radius or diameter of a circle from the length of a chord and the distance between the center of the circle and the chord. For example: In a circle, the length of a chord is 24 cm, and the distance between the center of the circle and the chord is 5 cm. Find the length of the radius (video). Finally, students will learn why two chords are congruent if and only if they're equidistant from the center (video).
Conclude by giving your students these challenges:
- Smaller and Smaller by NRICH
- Card Trick 2 by NRICH
- A Square in a Circle by NRICH
- 2005 AMC 8, Problem 10
- Triangles of Ten by Pierce School: Problem / Solution