2021 Fall AMC 12B Problems/Problem 22
Contents
Problem
Right triangle has side lengths , , and . A circle centered at is tangent to line at and passes through . A circle centered at is tangent to line at and passes through . What is ?
Diagram
Solution 1
Let be the midpoint of ; so . Let be the point such that is a rectangle. Then and . Let ; so . Then
Solution 2
This one uses the same diagram as Solution 1, except we draw . After doing angle chasing we find and , resulting in .
We also find that and , resulting in . .
-ThisUsernameIsTaken
Solution 3 (Analytic Geometry)
In a Cartesian plane, let and be respectively.
By analyzing the behaviors of the two circles, we set to be and be .
Hence derive the two equations:
Considering the coordinates of and for the two equations respectively, we get:
Solve to get and
Through using the distance formula,
.
~Wilhelm Z
Solution 4
Because the circle with center passes through points and and is tangent to line at point , is on the perpendicular bisector of segment and .
Because the circle with center passes through points and and is tangent to line at point , is on the perpendicular bisector of segment and .
Let lines and intersect at point . Hence, is a rectangle.
Denote by the midpoint of segment . Hence, . Because and are on the perpendicular bisector of segment , points , , are collinear with .
We have . Hence, . Hence, . Hence, .
We have . Hence, . Therefore, .
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 5
Let be the origin, making and . Let be the midpoint of ; .
Notice that both and must be on the perpendicular bisector of . The slope of is , making the 's slope be . Since passes through , the equation for becomes
using the slope intersect form. Since is perpendicular to and is perpendicular to (cause of tangencies), the -coordinate for is and the -coordinate for is . Plugging these numbers in the equation for gives and . Thus,
~ sml1809
Video Solution
~MathProblemSolvingSkills.com
Video Solution by Mathematical Dexterity
https://www.youtube.com/watch?v=ctx67nltpE0
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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All AMC 12 Problems and Solutions |
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