2020 AMC 10B Problems/Problem 8
Revision as of 06:28, 30 September 2021 by MRENTHUSIASM (talk | contribs) (→Solution 4 (Algebra): I found some flaws in this solution too. First, the equation was not solved correctly. Secondly, we should do casework which one is the right angle before we apply the Pythagorean Theorem.)
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)
~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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