2022 AMC 12A Problems/Problem 20
Problem
Isosceles trapezoid 
has parallel sides 
and 
with 
and 
There is a point 
in the plane such that 
and 
What is 
Solution 1
Consider the reflection 
of over the perpendicular bisector of 
, creating two new isosceles trapezoids 
and 
. Under this reflection, 
, 
, 
, and 
. By Ptolmey's theorem 
Thus 
and 
; dividing these two equations and taking the reciprocal yields 
.
Solution 2 (Cheese)
Notice that the question never says what the height of the trapezoid is; the only property we know about it is that 
. Therefore, we can say WLOG that the height of the trapezoid is 
and all 
points, including , lie on the same line with 
. Notice that this satisfies the problem requirements because 
, and 
.
Now all we have to find is 
~KingRavi
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 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 12 Problems and Solutions | |
| 2022 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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 | ||
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