Difference between revisions of "2021 Fall AMC 12B Problems/Problem 22"
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<cmath>OP=MP-MO=AM\cot\theta - BM\tan\theta = 5(\tfrac 43 - \tfrac 34) = \boxed{\textbf{(C)}\ \tfrac{35}{12}}.</cmath> | <cmath>OP=MP-MO=AM\cot\theta - BM\tan\theta = 5(\tfrac 43 - \tfrac 34) = \boxed{\textbf{(C)}\ \tfrac{35}{12}}.</cmath> | ||
− | ==Solution 2 (Analytic Geometry) == | + | == Solution 2== |
+ | |||
+ | This one uses the same diagram as Solution 1. After doing angle chasing we find <math>\triangle BPM \sim \triangle BAC</math> and <math>\frac{BM}{BC} = \frac{PM}{AC}</math>, resulting in <math>PM = \frac{20}{3}</math>. | ||
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+ | We also find that <math>\triangle BOM \sim \triangle ABC</math> and <math>\frac{BM}{AC} = \frac{OM}{BC}</math>, resulting in <math>OM = \frac{15}{4}</math>. <math>OP = PM - OM = \frac{35}{12}</math>. | ||
+ | |||
+ | ==Solution 3 (Analytic Geometry) == | ||
In a Cartesian plane, let <math>C, B,</math> and <math>A</math> be <math>(0,0),(0,6),(8,0)</math> respectively. | In a Cartesian plane, let <math>C, B,</math> and <math>A</math> be <math>(0,0),(0,6),(8,0)</math> respectively. |
Revision as of 09:17, 10 November 2022
Contents
[hide]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. After doing angle chasing we find and , resulting in .
We also find that and , resulting in . .
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 3
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)
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 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.