2021 CMC 12A Problems/Problem 6
- The following problem is from both the 2021 CMC 12A #6 and 2021 CMC 10A #7, so both problems redirect to this page.
Problem
How many of the following statements are true for every parallelogram ?
i. The perpendicular bisectors of the sides of all share at least one common point. ii. The perpendicular bisectors of the sides of are all distinct. iii. If the perpendicular bisectors of the sides of all share at least one common point then is a square. iv. If the perpendicular bisectors of the sides of are all distinct then these bisectors form a parallelogram.
Solution
Statement (i) is obviously false; suppose is a non-rectangular parallelogram, then the statement does not hold.
Statement (ii) is also false; suppose is a rectangle, then the perpendicular bisectors of opposite sides are the same.
Statement (iii) is also false; if we let be a non-square rectangle, the bisectors still share at least one common point at the center of the rectangle.
Statement (iv) is true. A parallelogram has two pairs of parallel opposite sides. Thus the perpendicular bisectors of those opposite sides will also be parallel, and thus if they are distinct, they will form a parallelogram.
In conclusion, only one statement is true, namely statement (iv), so the answer is .
See also
2021 CMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All CMC 12 Problems and Solutions |
2021 CMC 10A (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All CMC 10 Problems and Solutions |
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.