Difference between revisions of "2021 Fall AMC 12B Problems/Problem 10"
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<math>\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380</math> | <math>\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380</math> | ||
| − | ==Solution== | + | ==Solution 1 (Quick Look for Symmetry)== |
| − | + | ||
| + | By inspection, we may obtain the following choices for which symmetric isosceles triangles could be constructed within the unit circle described: | ||
| + | |||
| + | <math>20^\circ</math>, <math>50^\circ</math>, <math>80^\circ</math>, and <math>230^\circ</math>. | ||
| + | |||
| + | Thus we have <math>20+50+80+230=\boxed{(\textbf{E} )380}</math>. | ||
| + | |||
~Wilhelm Z | ~Wilhelm Z | ||
Revision as of 03:37, 24 November 2021
Problem
What is the sum of all possible values of 
between 
and 
such that the triangle in the coordinate plane whose vertices are 
, 
, and 
is isosceles?
Solution 1 (Quick Look for Symmetry)
By inspection, we may obtain the following choices for which symmetric isosceles triangles could be constructed within the unit circle described:
, 
, 
, and 
.
Thus we have 
.
~Wilhelm Z
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
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| All AMC 12 Problems and Solutions | |
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