Difference between revisions of "1995 AIME Problems/Problem 9"

Latest revision as of 12:47, 3 February 2023

Problem

Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$

$[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(1,1.63),W); dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]$

Solution 1

Let $x=\angle CAM$ , so $3x=\angle CDM$ . Then, $\frac{\tan 3x}{\tan x}=\frac{CM/1}{CM/11}=11$ . Expanding $\tan 3x$ using the angle sum identity gives $\tan 3x=\tan(2x+x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.$ Thus, $\frac{3-\tan^2x}{1-3\tan^2x}=11$ . Solving, we get $\tan x= \frac 12$ . Hence, $CM=\frac{11}2$ and $AC= \frac{11\sqrt{5}}2$ by the Pythagorean Theorem. The total perimeter is $2(AC + CM) = \sqrt{605}+11$ . The answer is thus $a+b=\boxed{616}$ .

Solution 2

In a similar fashion, we encode the angles as complex numbers, so if $BM=x$ , then $\angle BAD=\text{Arg}(11+xi)$ and $\angle BDM=\text{Arg}(1+xi)$ . So we need only find $x$ such that $\text{Arg}((11+xi)^3)=\text{Arg}(1331-33x^2+(363x-x^3)i)=\text{Arg}(1+xi)$ . This will happen when $\frac{363x-x^3}{1331-33x^2}=x$ , which simplifies to $121x-4x^3=0$ . Therefore, $x=\frac{11}{2}$ . By the Pythagorean Theorem, $AB=\frac{11\sqrt{5}}{2}$ , so the perimeter is $11+11\sqrt{5}=11+\sqrt{605}$ , giving us our answer, $\boxed{616}$ .

Solution 3

Let $\angle BAD=\alpha$ , so $\angle BDM=3\alpha$ , $\angle BDA=180-3\alpha$ , and thus $\angle ABD=2\alpha.$ We can then draw the angle bisector of $\angle ABD$ , and let it intersect $\overline{AM}$ at $E.$ Since $\angle BAE=\angle ABE$ , $AE=BE.$ Let $AE=x$ . Then we see by the Pythagorean Theorem, $BM=\sqrt{BE^2-ME^2}=\sqrt{x^2-(11-x)^2}=\sqrt{22x-121}$ , $BD=\sqrt{BM^2+1}=\sqrt{22x-120}$ , $BA=\sqrt{BM^2+121}=\sqrt{22x}$ , and $DE=10-x.$ By the angle bisector theorem, $BA/BD=EA/ED.$ Substituting in what we know for the lengths of those segments, we see that $\frac{\sqrt{22x}}{\sqrt{22x-120}}=\frac{x}{10-x}.$ multiplying by both denominators and squaring both sides yields $22x(10-x)^2=x^2(22x-120)$ which simplifies to $x=\frac{55}{8}.$ Substituting this in for x in the equations for $BA$ and $BM$ yields $BA=\frac{\sqrt{605}}{2}$ and $BM=\frac{11}{2}.$ Thus the perimeter is $11+\sqrt{605}$ , and the answer is $\boxed{616}$ .

Solution 4

The triangle is symmetrical so we can split it in half ( $\triangle ABM$ and $\triangle ACM$ ).

Let $\angle BAM = y$ and $\angle BDM = 3y$ . By the Law of Sines on triangle $BAD$ , $\frac{10}{\sin 2y} = \frac{BD}{\sin y}$ . Using $\sin 2y = 2\sin y\cos y$ we can get $BD = \frac{5}{\cos y}$ . We can use this information to relate $BD$ to $DM$ by using the Law of Sines on triangle $BMD$ .

$\frac{\frac{5}{\cos y}}{\sin BMD} = \frac{1}{\sin 90^\circ - 3y}$

$\sin BMD = 1$ (as $\angle BMD$ is a right angle), so $\frac{1}{\sin 90^\circ - 3y} = \frac{5}{\cos y}$ . Using the identity $\sin 90^\circ - x = \cos x$ , we can turn the equation into::

$\frac{1}{\cos 3y} = \frac{5}{\cos y}$

$5\cos 3y = \cos y$

$5(4\cos ^3 y - 3\cos y) = \cos y$

$20\cos ^3 y = 16 \cos y$

$5\cos ^3 y = 4\cos y$

$5\cos ^2 y = 4$

$\cos ^2 y = \frac{4}{5}$

Now that we've found $\cos y$ , we can look at the side lengths of $BM$ and $AB$ (since they are symmetrical, the perimeter of $\triangle ABC$ is $2(BM + AB)$ .

We note that $BM = 11\tan y$ and $AB = 11\sec y$ .

$\sin ^2 y = 1 - \cos ^2 y$

$\sin ^2 y = \frac{1}{5}$

$\tan ^2 y = \frac{1}{4}$

$\tan y = \frac{1}{2}$

(Note it is positive since $BM > 0$ ).

$\sec ^2 y = \frac{5}{4}$

$\sec y = \frac{\sqrt{5}}{2}$

$BM + AB = 11\frac{\sqrt{5}+1}{2}$

$2(BM + AB) = 11(\sqrt{5} + 1)$

$2(BM + AB) = 11\sqrt{5} + 11$

$2(BM + AB) = \sqrt{605} + 11$

The answer is $\boxed{616}$ .

Solution 5

Suppose $\angle BAM=\angle CAM =x$ , since $\angle BDC=3\angle BAC$ , we have $\angle BDM=\angle MDC = 3x$ . Therefore, $\angle DBC=\angle DCB = 90^\circ -3x$ and $\angle ABD=\angle DCA=2x$ . As a result, $\triangle KAC$ is isosceles, $KC=KA$ .

Let $H$ be a point on the extension of $CD$ through $D$ such that $\overline{HB}\perp\overline{BC}$ and denote the intersection of $\overline{HC}$ and $\overline{AB}$ as $K$ . Then, $BH=2DM=2, \overline{HB}\parallel\overline{DM}$ , and $HD=DC$ by the Midpoint Theorem. So, $\angle HBA=x$ and $\angle CDM=\angle CHB=\angle HDA= 3x$ .

Consequently, $\triangle HBK\sim \triangle DAK$ , $\frac{BK}{KA}=\frac{HK}{KD}=\frac{1}{5}$ Assume $BK=a$ and $HK=b$ , then $KA=5a$ and $KD = 5b$ . Since $KC=KA, KC=5a$ , and since $HD=DC$ , $KC=11b$ . Therefore, $a=\frac{11}{5}b$ .

In $\triangle BDM$ , by the Pythagorean Theorem, $BM=\sqrt{36b^2-1}$ . Similarly in $\triangle BAM$ , $BM=\sqrt{36a^2-121}$ . So $\sqrt{36a^2-121}=\sqrt{36b^2-1}$ Since $a=\frac{11}{5}b$ , we have $b=\frac{5\sqrt{5}}{12}$ and $a=\frac{11\sqrt{5}}{12}$ . Consequently, $BM=\frac{11}{2}$ and $AB=\frac{11\sqrt{5}}{2}$ . Thus, the perimeter of $\triangle ABC$ is $11+\sqrt{605}$ , and the answer is $\boxed{616}$ .

@@ Line 1: / Line 1: @@
 == Problem ==
+Triangle <math>ABC</math> is [[isosceles triangle|isosceles]], with <math>AB=AC</math> and [[altitude]] <math>AM=11.</math>  Suppose that there is a point <math>D</math> on <math>\overline{AM}</math> with <math>AD=10</math> and <math>\angle BDC=3\angle BAC.</math>  Then the perimeter of <math>\triangle ABC</math> may be written in the form <math>a+\sqrt{b},</math> where <math>a</math> and <math>b</math> are integers.  Find <math>a+b.</math>
-== Solution ==
+<asy>
+import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3;
+draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(1,1.63),W);
+dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE);
+clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);</asy>
+<!-- [[Image:AIME_1995_Problem_9.png|center]] -->
+== Solution 1 ==
+Let <math>x=\angle CAM</math>, so <math>3x=\angle CDM</math>. Then, <math>\frac{\tan 3x}{\tan x}=\frac{CM/1}{CM/11}=11</math>. Expanding <math>\tan 3x</math> using the angle sum identity gives <cmath>\tan 3x=\tan(2x+x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.</cmath>
+Thus, <math>\frac{3-\tan^2x}{1-3\tan^2x}=11</math>. Solving, we get <math>\tan x= \frac 12</math>. Hence, <math>CM=\frac{11}2</math> and <math>AC= \frac{11\sqrt{5}}2</math> by the [[Pythagorean Theorem]]. The total perimeter is <math>2(AC + CM) = \sqrt{605}+11</math>. The answer is thus <math>a+b=\boxed{616}</math>.
+== Solution 2 ==
+In a similar fashion, we encode the angles as complex numbers, so if <math>BM=x</math>, then <math>\angle BAD=\text{Arg}(11+xi)</math> and <math>\angle BDM=\text{Arg}(1+xi)</math>. So we need only find <math>x</math> such that <math>\text{Arg}((11+xi)^3)=\text{Arg}(1331-33x^2+(363x-x^3)i)=\text{Arg}(1+xi)</math>. This will happen when <math>\frac{363x-x^3}{1331-33x^2}=x</math>,  which simplifies to <math>121x-4x^3=0</math>. Therefore, <math>x=\frac{11}{2}</math>. By the Pythagorean Theorem, <math>AB=\frac{11\sqrt{5}}{2}</math>, so the perimeter is <math>11+11\sqrt{5}=11+\sqrt{605}</math>, giving us our answer, <math>\boxed{616}</math>.
+== Solution 3 ==
+Let <math>\angle BAD=\alpha</math>, so <math>\angle BDM=3\alpha</math>, <math>\angle BDA=180-3\alpha</math>, and thus <math>\angle ABD=2\alpha.</math> We can then draw the angle bisector of <math>\angle ABD</math>, and let it intersect <math>\overline{AM}</math> at <math>E.</math> Since <math>\angle BAE=\angle ABE</math>, <math>AE=BE.</math> Let <math>AE=x</math>. Then we see by the Pythagorean Theorem, <math>BM=\sqrt{BE^2-ME^2}=\sqrt{x^2-(11-x)^2}=\sqrt{22x-121}</math>, <math>BD=\sqrt{BM^2+1}=\sqrt{22x-120}</math>, <math>BA=\sqrt{BM^2+121}=\sqrt{22x}</math>, and <math>DE=10-x.</math> By the angle bisector theorem, <math>BA/BD=EA/ED.</math> Substituting in what we know for the lengths of those segments, we see that <cmath>\frac{\sqrt{22x}}{\sqrt{22x-120}}=\frac{x}{10-x}.</cmath> multiplying by both denominators and squaring both sides yields <cmath>22x(10-x)^2=x^2(22x-120)</cmath> which simplifies to <math>x=\frac{55}{8}.</math> Substituting this in for x in the equations for <math>BA</math> and <math>BM</math> yields <math>BA=\frac{\sqrt{605}}{2}</math> and <math>BM=\frac{11}{2}.</math> Thus the perimeter is <math>11+\sqrt{605}</math>, and the answer is <math>\boxed{616}</math>.
+== Solution 4 ==
+The triangle is symmetrical so we can split it in half (<math>\triangle ABM</math> and <math>\triangle ACM</math>).
+Let <math>\angle BAM = y</math> and <math>\angle BDM = 3y</math>. By the Law of Sines on triangle <math>BAD</math>, <math>\frac{10}{\sin 2y} = \frac{BD}{\sin y}</math>. Using <math>\sin 2y = 2\sin y\cos y</math> we can get <math>BD = \frac{5}{\cos y}</math>. We can use this information to relate <math>BD</math> to <math>DM</math> by using the Law of Sines on triangle <math>BMD</math>.
+<cmath>\frac{\frac{5}{\cos y}}{\sin BMD} = \frac{1}{\sin 90^\circ - 3y}</cmath>
+<math>\sin BMD = 1</math> (as <math>\angle BMD</math> is a right angle), so <math>\frac{1}{\sin 90^\circ - 3y} = \frac{5}{\cos y}</math>. Using the identity <math>\sin 90^\circ - x = \cos x</math>, we can turn the equation into::
+<cmath>\frac{1}{\cos 3y} = \frac{5}{\cos y}</cmath>
+<cmath>5\cos 3y = \cos y</cmath>
+<cmath>5(4\cos ^3 y - 3\cos y) = \cos y</cmath>
+<cmath>20\cos ^3 y = 16 \cos y</cmath>
+<cmath>5\cos ^3 y = 4\cos y</cmath>
+<cmath>5\cos ^2 y = 4</cmath>
+<cmath>\cos ^2 y = \frac{4}{5}</cmath>
+Now that we've found <math>\cos y</math>, we can look at the side lengths of <math>BM</math> and <math>AB</math> (since they are symmetrical, the perimeter of <math>\triangle ABC</math> is <math>2(BM + AB)</math>.
+We note that <math>BM = 11\tan y</math> and <math>AB = 11\sec y</math>.
+<cmath>\sin ^2 y = 1 - \cos ^2 y</cmath>
+<cmath>\sin ^2 y = \frac{1}{5}</cmath>
+<cmath>\tan ^2 y = \frac{1}{4}</cmath>
+<cmath>\tan y = \frac{1}{2}</cmath>
+(Note it is positive since <math>BM > 0</math>).
+<cmath>\sec ^2 y = \frac{5}{4}</cmath>
+<cmath>\sec y = \frac{\sqrt{5}}{2}</cmath>
+<cmath>BM + AB = 11\frac{\sqrt{5}+1}{2}</cmath>
+<cmath>2(BM + AB) = 11(\sqrt{5} + 1)</cmath>
+<cmath>2(BM + AB) = 11\sqrt{5} + 11</cmath>
+<cmath>2(BM + AB) = \sqrt{605} + 11</cmath>
+The answer is <math>\boxed{616}</math>.
+== Solution 5 ==
+Suppose <math>\angle BAM=\angle CAM =x</math>, since <math>\angle BDC=3\angle BAC</math>, we have <math>\angle BDM=\angle MDC = 3x</math>. Therefore, <math>\angle DBC=\angle DCB = 90^\circ -3x</math> and <math>\angle ABD=\angle DCA=2x</math>. As a result, <math>\triangle KAC</math> is isosceles, <math>KC=KA</math>.
+Let <math>H</math> be a point on the extension of <math>CD</math> through <math>D</math> such that <math>\overline{HB}\perp\overline{BC}</math> and denote the intersection of <math>\overline{HC}</math> and <math>\overline{AB}</math> as <math>K</math>. Then, <math>BH=2DM=2, \overline{HB}\parallel\overline{DM}</math>, and <math>HD=DC</math> by the [[Midpoint]] Theorem. So, <math>\angle HBA=x</math> and <math>\angle CDM=\angle CHB=\angle HDA= 3x</math>.
+Consequently, <math>\triangle HBK\sim \triangle DAK</math>, <cmath>\frac{BK}{KA}=\frac{HK}{KD}=\frac{1}{5}</cmath> Assume <math>BK=a</math> and <math>HK=b</math>, then <math>KA=5a</math> and <math>KD = 5b</math>. Since <math>KC=KA, KC=5a</math>, and since <math>HD=DC</math>, <math>KC=11b</math>. Therefore, <math>a=\frac{11}{5}b</math>.
+In <math>\triangle BDM</math>, by the [[Pythagorean Theorem]], <math>BM=\sqrt{36b^2-1}</math>. Similarly in <math>\triangle BAM</math>, <math>BM=\sqrt{36a^2-121}</math>. So <cmath>\sqrt{36a^2-121}=\sqrt{36b^2-1}</cmath> Since <math>a=\frac{11}{5}b</math>, we have <math>b=\frac{5\sqrt{5}}{12}</math> and <math>a=\frac{11\sqrt{5}}{12}</math>. Consequently, <math>BM=\frac{11}{2}</math> and <math>AB=\frac{11\sqrt{5}}{2}</math>. Thus, the perimeter of <math>\triangle ABC</math> is <math>11+\sqrt{605}</math>, and the answer is <math>\boxed{616}</math>.
 == See also ==
-* [[1995 AIME Problems]]
+{{AIME box|year=1995|num-b=8|num-a=10}}
+[[Category:Intermediate Geometry Problems]]
+[[Category:Intermediate Trigonometry Problems]]
+{{MAA Notice}}

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