Difference between revisions of "2020 AIME I Problems/Problem 1"

(Solution)
(Solution)
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== Solution ==
 
== Solution ==
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<asy>
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size(10cm);
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pair A, B, C, D, F;
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A = (0, tan(3 * pi / 7));
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B = (1, 0);
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C = (-1, 0);
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F = rotate(90/7, A) * (A - (0, 2));
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D = rotate(900/7, F) * A;
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draw(A -- B -- C -- cycle);
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draw(F -- D);
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draw(D -- B);
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label("$A$", A, W);
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label("$B$", B, E);
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label("$C$", C, W);
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label("$D$", D, W);
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label("$E$", F, E);
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</asy>
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If we set <math>\angle{BAC}</math> to <math>x</math>, we can find all other angles through these two properties:
 
If we set <math>\angle{BAC}</math> to <math>x</math>, we can find all other angles through these two properties:
 
1. Angles in a triangle sum to <math>180^{\circ}</math>.
 
1. Angles in a triangle sum to <math>180^{\circ}</math>.

Revision as of 00:03, 13 March 2020

Problem

In $\triangle ABC$ with $AB=AC,$ point $D$ lies strictly between $A$ and $C$ on side $\overline{AC},$ and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC.$ The degree measure of $\angle ABC$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

[asy] size(10cm); pair A, B, C, D, F; A = (0, tan(3 * pi / 7)); B = (1, 0); C = (-1, 0); F = rotate(90/7, A) * (A - (0, 2)); D = rotate(900/7, F) * A;  draw(A -- B -- C -- cycle); draw(F -- D); draw(D -- B);  label("$A$", A, W); label("$B$", B, E); label("$C$", C, W); label("$D$", D, W); label("$E$", F, E); [/asy]

If we set $\angle{BAC}$ to $x$, we can find all other angles through these two properties: 1. Angles in a triangle sum to $180^{\circ}$. 2. The base angles of an isoceles triangle are congruent.

Now we angle chase. $\angle{ADE}=\angle{EAD}=x$, $\angle{AED} = 180-2x$, $\angle{BED}=\angle{EBD}=2x$, $\angle{EDB} = 180-4x$, $\angle{BDC} = \angle{BCD} = 3x$, $\angle{CBD} = 180-6x$. Since $AB = AC$ as given by the problem, $\angle{ABC} = \angle{ACB}$, so $180-4x=3x$. Therefore, $x = 180/7^{\circ}$, and our desired angle is \[180-4\left(\frac{180}{7}\right) = \frac{540}{7}\] for an answer of $\boxed{547}$.

-molocyxu

See Also

2020 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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