Difference between revisions of "2021 Fall AMC 12B Problems/Problem 22"
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<math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ | <math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ | ||
\frac{73}{25} \qquad\textbf{(E)}\ 3</math> | \frac{73}{25} \qquad\textbf{(E)}\ 3</math> | ||
+ | |||
+ | ==Diagram== | ||
+ | <center><asy> | ||
+ | defaultpen(fontsize(11)+0.8); size(200); | ||
+ | pair A,B,C,M,Ic,Ib,O,P; | ||
+ | C=MP("C",origin,down+left); A=MP("A",8*right,down+right); B=MP("B",6*up,2*up); draw(A--B--C--A); draw(B--(B+A), gray+0.25); M=MP("M",(A+B)/2,down+left); O=MP("O",extension(B,B+A,M,M+(B-M)*dir(-90)),down); P=MP("P",extension(A,B+A,M,M+(B-M)*dir(-90)),up); draw(M--P^^A--P, gray+0.25); label("$\theta$", A, 7*dir(162)); label("$\theta$", B, 7*dir(-20)); label("$\theta$", P, 7*dir(-110)); label("$6$", B--C, left); label("$8$", A--C, down); | ||
+ | </asy></center> | ||
+ | |||
==Solution 1 (Analytic Geometry) == | ==Solution 1 (Analytic Geometry) == |
Revision as of 17:17, 6 June 2022
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 (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 2
This solution is based on this figure: Image:2021_AMC_12B_(Nov)_Problem_22,_sol.png.
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 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.