2020 AMC 10B Problems/Problem 8
Contents
Problem
Points and lie in a plane with . How many locations for point in this plane are there such that the triangle with vertices , , and is a right triangle with area square units?
Solution 1 (Geometry)
Let the brackets denote areas. We are given that Since it follows that
We construct a circle with diameter All such locations for are shown below:
We apply casework to the right angle of
 If then by the tangent.
 If then by the tangent.
 If then by the Inscribed Angle Theorem.
Together, there are such locations for
Remarks
 The reflections of about are respectively.
 The reflections of about the perpendicular bisector of are respectively.
~MRENTHUSIASM
Solution 2 (Algebra)
Let the brackets denote areas. We are given that Since it follows that
Without the loss of generality, let and We conclude that the coordinate of must be
We apply casework to the right angle of

The coordinate of must be so we have
In this case, there are such locations for

The coordinate of must be so we have
In this case, there are such locations for

For the Pythagorean Theorem gives Solving this equation, we have or
For we have by a similar process.
In this case, there are such locations for
Together, there are such locations for
~MRENTHUSIASM ~mewto
Video Solution (HOW TO CRITICALLY THINK!!!)
~Education, the Study of Everything
Video Solution
~savannahsolver
Video Solution
https://youtu.be/GrCtzL0SUo?t=19
See Also
2020 AMC 10B (Problems • Answer Key • Resources)  
Preceded by Problem 7 
Followed by Problem 9  
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.