Difference between revisions of "2015 AIME I Problems/Problem 4"

Revision as of 21:21, 6 September 2022

Problem

Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$ . Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$ . Let $M$ be the midpoint of $\overline{AE}$ , and $N$ be the midpoint of $\overline{CD}$ . The area of $\triangle BMN$ is $x$ . Find $x^2$ .

Diagram

$[asy] pair A = (0, 0), B = (16, 0), C = (20, 0), D = (8, 8*sqrt(3)), EE = (18, 2*sqrt(3)), M = (9, sqrt(3)), NN = (14, 4*sqrt(3)); draw(A--B--D--cycle); draw(B--C--EE--cycle); draw(A--EE); draw(C--D); draw(B--M--NN--cycle); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(M); dot(NN); label("A", A, SW); label("B", B, S); label("C", C, SE); label("D", D, N); label("E", EE, N); label("M", M, NW); label("N", NN, NE); [/asy]$

Diagram by RedFireTruck (talk) 18:52, 15 February 2021 (EST)

Solution 1

Let $A$ be the origin, so $B=(16,0)$ and $C=(20,0).$ Using equilateral triangle properties tells us that $D=(8,8\sqrt3)$ and $E=(18,2\sqrt3)$ as well. Therefore, $M=(9,\sqrt3)$ and $N=(14,4\sqrt3).$ Applying the Shoelace Theorem to triangle $BMN$ gives

$x=\dfrac 1 2 |16\sqrt3+36\sqrt3+0-(0+14\sqrt3+64\sqrt3)| =13\sqrt3,$

so $x^2=\boxed{507}.$

Solution 2

Note that $AB=DB=16$ and $BE=BC=4$ . Also, $\angle ABE = \angle DBC = 120^{\circ}$ . Thus, $\triangle ABE \cong \triangle DBC$ by SAS.

From this, it is clear that a $60^{\circ}$ rotation about $B$ will map $\triangle ABE$ to $\triangle DBC$ . This rotation also maps $M$ to $N$ . Thus, $BM=BN$ and $\angle MBN=60^{\circ}$ . Thus, $\triangle BMN$ is equilateral.

Using the Law of Cosines on $\triangle ABE$ , $AE^2 = 16^2 + 4^2 - 2\cdot 16\cdot 4\cdot\left(-\frac{1}{2}\right)$ $AE = 4\sqrt{21}$ Thus, $AM=ME=2\sqrt{21}$ .

Using Stewart's Theorem on $\triangle ABE$ , $AM\cdot ME\cdot AE + AE\cdot BM^2 = BE^2\cdot AM + BA^2\cdot ME$ $BM = 2\sqrt{13}$

Calculating the area of $\triangle BMN$ , $[BMN] = \frac{\sqrt{3}}{4} BM^2$ $[BMN] = 13\sqrt{3}$ Thus, $x=13\sqrt{3}$ , so $x^2 = 507$ . Our final answer is $\boxed{507}$ .

Admittedly, this is much more tedious than the coordinate solutions.

I also noticed that there are two more ways of showing that $\triangle BMN$ is equilateral:

One way is to show that $\triangle ADB$ , $\triangle BMN$ , and $\triangle ECB$ are related by a spiral similarity centered at $B$ .

The other way is to use the Mean Geometry Theorem. Note that $\triangle BCE$ and $\triangle BDA$ are similar and have the same orientation. Note that $B$ is the weighted average of $B$ and $B$ , $M$ is the weighted average of $E$ and $A$ , and $N$ is the weighted average of $C$ and $D$ . The weights are the same for all three averages. (The weights are actually just $\frac{1}{2}$ and $\frac{1}{2}$ , so these are also unweighted averages.) Thus, by the Mean Geometry Theorem, $\triangle BMN$ is similar to both $\triangle BAD$ and $\triangle BEC$ , which means that $\triangle BMN$ is equilateral.

Solution 3

$AB = BD, BE = BC, \angle ABE = \angle CBD \implies \triangle ABE= \triangle DBC$

Medians are equal, so $MB = MN, \angle ABM = \angle DBN \implies$ $\angle MBN = \angle ABD - \angle ABM + \angle DBN = 60^\circ \implies$

$\triangle MNB$ is equilateral triangle.

The height of $\triangle BCE$ is $2 \sqrt{3},$ distance from $A$ to midpoint $BC$ is $16 + 2 = 18 \implies \frac {AE^2}{4} =\frac{ (16 + 2)^2 +2^2 \cdot 3}{4} = 81 + 3 = 84.$

$BM$ is the median of $\triangle ABE \implies$ $BM^2 = \frac {AB^2}{2} + \frac {BE^2}{2} - \frac {AE^2}{4}=16 \cdot 8 + 4 \cdot 2 - 84 = 52.$

The area of $\triangle BMN$

$[BMN] = \frac{\sqrt{3}}{4} BM^2 =13 \sqrt{3} \implies \boxed{\textbf{507}}.$

vladimir.shelomovskii@gmail.com, vvsss

@@ Line 28: / Line 28: @@
 Diagram by [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FFFFFF">talk</font>]]) 18:52, 15 February 2021 (EST)
-==Solution 1 (fastest)==
+==Solution 1==
-Let point <math>A</math> be at <math>(0,0)</math>. Then, <math>B</math> is at <math>(16,0)</math>, and <math>C</math> is at <math>(20,0)</math>. Due to symmetry, it is allowed to assume <math>D</math> and <math>E</math> are in quadrant 1. By equilateral triangle calculations, Point <math>D</math> is at <math>(8,8\sqrt{3})</math>, and Point <math>E</math> is at <math>(18,2\sqrt{3})</math>. By Midpoint Formula, <math>M</math> is at <math>(9,\sqrt{3})</math>, and <math>N</math> is at <math>(14,4\sqrt{3})</math>. The distance formula shows that <math>BM=BN=MN=2\sqrt{13}</math>. Therefore, by equilateral triangle area formula <math>\textbf{OR}</math> by Shoelace Theorem, <math>x=13\sqrt{3}</math>, so <math>x^2</math> is <math>\boxed{507}</math>.
+Let <math>A</math> be the origin, so <math>B=(16,0)</math> and <math>C=(20,0).</math> Using equilateral triangle properties tells us that <math>D=(8,8\sqrt3)</math> and <math>E=(18,2\sqrt3)</math> as well. Therefore, <math>M=(9,\sqrt3)</math> and <math>N=(14,4\sqrt3).</math> Applying the Shoelace Theorem to triangle <math>BMN</math> gives
+<cmath>x=\dfrac 1 2 |16\sqrt3+36\sqrt3+0-(0+14\sqrt3+64\sqrt3)| =13\sqrt3,</cmath>
+so <math>x^2=\boxed{507}.</math>
 ==Solution 2==

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