2020 AMC 10B Problems/Problem 8
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
~IceMatrix
~savannahsolver
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 |
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