Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 CMC 12A Problems/Problem 6"

(Created page with "{{duplicate| 2021 CMC 12A #6 and 2021 CMC 10A #7}} ==Problem== How many of the following statements are true for every...")
 
(whoops – forgot the A)
 
Line 23: Line 23:
  
 
==See also==
 
==See also==
−
{{CMC12 box|year=2021|num-b=5|num-a=7}}
+
{{CMC12 box|year=2021|ab=A|num-b=5|num-a=7}}
−
{{CMC10 box|year=2021|num-b=6|num-a=8}}
+
{{CMC10 box|year=2021|ab=A|num-b=6|num-a=8}}
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 
{{MAC Notice}}
 
{{MAC Notice}}

Latest revision as of 12:49, 4 January 2021

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 $\mathcal{P}$? 

      i. The perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point.
      ii. The perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct.
      iii. If the perpendicular bisectors of the sides of $\mathcal{P}$ all share at least one common point then $\mathcal{P}$ is a square.
      iv. If the perpendicular bisectors of the sides of $\mathcal{P}$ are all distinct then these bisectors form a parallelogram.

$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4\qquad$

Solution

Statement (i) is obviously false; suppose $\mathcal{P}$ is a non-rectangular parallelogram, then the statement does not hold.

Statement (ii) is also false; suppose $\mathcal{P}$ is a rectangle, then the perpendicular bisectors of opposite sides are the same.

Statement (iii) is also false; if we let $\mathcal{P}$ 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 $\boxed{\textbf{(B) } 1}$.

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_CMC_12A_Problems/Problem_6&oldid=141501"