2022 AMC 12A Problems/Problem 20
Contents
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)
Video Solution by OmegaLearn
~ pi_is_3.14
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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