Students will be asked to prove the diagonals of a kite are perpendicular. If students can't prove it, show them this proof. Then have students use the theorem to solve some basic problems. Here's an example.
Conclude by giving your students these challenges:
- Stars by NRICH
- Treasure Hunt by NRICH
- 2007 AMC 8, Problem 2
Follow the Numbers by NRICH: Order of digits doesn't matter, their initial sum will be the same. For example, 73 and 37 have the same initial sum. In fact, many numbers will have the same initial sum. For example, 4, 13, 22, 31, and 40, all have the same initial sum. From this observation, we can figure out where the numbers 1-100 go by following 18 possibilities, because 1 has the smallest initial sum, which is 1, and 99 has the largest initial sum, which is 18. Some of these 18 will have the same journey, for the reasons previously stated. These are: 1 and 10, 2 and 11, ..., 9 and 18. We also know where a number goes once we find a number we've seen before. Following each journey in turn, remembering that our 1-18 are the initial sums, we have
- 1, 2, 4, 8, 16, 14, 10, ...
- 3, 6, 12, ...
- 5, 10, ...
- 18, 36, 18, ...
Thus, every number gets caught in 1 of 3 patterns:
- 2, 4, 8, 16, 14, 10, 2, ...
- 6, 12, 6, ...
- 18, 18, ...
This strategy can be used to find the patterns for the other two proposed rules as well:
- Add the digits, then multiply by 3.
- Add the digits, then multiply by 5.