Difference between revisions of "2021 Fall AMC 12B Problems/Problem 24"
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− | + | ==Solution== | |
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− | Because of this similarity, we have that <cmath>\frac{AC}{AD} = \frac{AE}{AB | + | <b>Claim:</b> <math>\triangle ADC \sim \triangle ABE.</math> |
+ | <b>Proof:</b> Note that <math>\angle CAD = \angle CAE = \angle EAB</math> and <math>\angle DCA = \angle BCA = \angle BEA</math> meaning that our claim is true by AA similarity. | ||
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+ | Because of this similarity, we have that <cmath>\frac{AC}{AD} = \frac{AE}{AB} \to AB \cdot AC = AD \cdot AE = AB \cdot AF</cmath> by Power of a Point. Thus, <math>AC=AF=20.</math> | ||
Now, note that <math>\angle CAF = \angle CAB</math> and plug into Law of Cosines to find the angle's cosine: <cmath>AB^2+AC^2-2\cdot AB \cdot AC \cdot \cos(\angle CAB) = BC^2 \to \cos(\angle CAB = -\frac{1}{8}.</cmath> | Now, note that <math>\angle CAF = \angle CAB</math> and plug into Law of Cosines to find the angle's cosine: <cmath>AB^2+AC^2-2\cdot AB \cdot AC \cdot \cos(\angle CAB) = BC^2 \to \cos(\angle CAB = -\frac{1}{8}.</cmath> |
Revision as of 17:52, 25 November 2021
Solution
Claim: Proof: Note that and meaning that our claim is true by AA similarity.
Because of this similarity, we have that by Power of a Point. Thus,
Now, note that and plug into Law of Cosines to find the angle's cosine:
So, we observe that we can use Law of Cosines again to find :
- kevinmathz