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 "2019 AMC 10A Problems/Problem 6"

m (→‎Solution)
(→‎Solution)
Line 12: Line 12:
 
==Solution==
 
==Solution==
 
This question is simply asking how many of the listed quadrilaterals are cyclic. A square, a rectangle, and an isosceles trapezoid (that isn't a parallelogram) are all cyclic, and the other two are not. Thus, the answer is <math>3 \implies \boxed{C}.</math>
 
This question is simply asking how many of the listed quadrilaterals are cyclic. A square, a rectangle, and an isosceles trapezoid (that isn't a parallelogram) are all cyclic, and the other two are not. Thus, the answer is <math>3 \implies \boxed{C}.</math>
−
 
−
Notice that the lines drawn can be represented by radii of a circle, which are all equidistant from its center <math>O</math>. From this, we can see that point <math>O</math> does not necessarily need to be inside the quadrilateral. Since the quadrilateral must have all its points on Circle <math>O</math>, it is a cyclic quadrilateral. Using the fact that opposite angles sum to <math>180^{\circ}</math>, the only quadrilaterals that fit this description are the square, the rectangle, and the isosceles trapezoid <math>\rightarrow \boxed{C}</math>.
 
  
 
==See Also==
 
==See Also==

Revision as of 17:23, 9 February 2019

Problem

For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?

  • a square
  • a rectangle that is not a square
  • a rhombus that is not a square
  • a parallelogram that is not a rectangle or a rhombus
  • an isosceles trapezoid that is not a parallelogram

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$

Solution

This question is simply asking how many of the listed quadrilaterals are cyclic. A square, a rectangle, and an isosceles trapezoid (that isn't a parallelogram) are all cyclic, and the other two are not. Thus, the answer is $3 \implies \boxed{C}.$

See Also

2019 AMC 10A (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 AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10A_Problems/Problem_6&oldid=101510"