The A-level textbook we use has a nice picture of the circumcircle of a triangle and a definition, plus a brief description of how to work through them. For those who are pondering what a circumcircle is, click on the image or link below
Image credit: WolframMathWorld
I’ll just stick to basic vocabulary in this post, rather than the formal circumcentre and circumradius.
Back to the book – not exactly inspiring or memorable stuff!
I looked at the class and off the cuff changed the lesson plan.
- Plain paper
Draw a decent size triangle on the paper. Label the corners A,B,C.
Using geometrical constructions, find the centre of the circle that your triangle fits in. Check by actually drawing the circle
Discuss what techniques gave the best results – hopefully you’ll have perpendicular bisectors. There is a nice comparison between bisecting the angles (which some students will do) and bisecting the sides. The angle bisectors always cross inside the triangle, the side bisectors don’t.
Randomly generate co-ordinates for A, B, & C. Get the students to pick them and then they can’t moan if the calculations are awful.
Discuss how you are going to find the centre and radius of the circumcircle. We decided on:
- Only use two sides
- Find the midpoints
- Find the gradients and hence perpendicular gradients
- Generate the equations of the lines through the midpoint
- Find where they intersect
- Use the point and one corner to find the radius
Review their methods, looking for premature rounding in questions. I’m still instilling an appreciation for the accuracy of fractions and surds, over reaching for the calculator.
This is how my solution looked – I numbered the picture and the steps so students could follow the logic. I was answering on one page projected on screen.