Here's what I did:
- Have students draw square ABCD that is at least 5 x 5 on graph paper.
- Draw diagonal BD
- Place four points anywhere along BD and label them F,G,H, and I (in order)
- Measure the distance from each point to the opposite corners A & C (so there are 8 lengths total)
- Make conclusion
The goal here was to kind of back in to the notion of angle bisectors, but all this part of the activity does is conclude that points along the diagonal of a square are equidistant from the corners. How can you show that the angles are in fact equal? That's where I got stuck.
So I very briefly addressed the idea of triangle congruence using SSS. If the sides are all the same, then the triangles have to be the same, and if the triangles are the same, then the angles have to be the same. But in actuality, that only deals with corresponding angles - I'm pretty sure you'd need to dig deeper, into the Triangle Sum Theorem to prove that each angle was 45 deg and bisected from the corner of the square.
No school tomorrow (Election Day), so I'll try to think of some way to tie it all together so we can close out Unit 3 with angle bisector construction (pray for me) and symmetry.