Difference between revisions of "1995 AIME Problems/Problem 9"
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− | == Solution == | + | == 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> | 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>. | 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>. | ||
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+ | == 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>. | ||
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+ | == 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{BM^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>. | ||
== See also == | == See also == | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
[[Category:Intermediate Trigonometry Problems]] | [[Category:Intermediate Trigonometry Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 00:51, 28 November 2015
Problem
Triangle is isosceles, with and altitude Suppose that there is a point on with and Then the perimeter of may be written in the form where and are integers. Find
Solution 1
Let , so . Then, . Expanding using the angle sum identity gives Thus, . Solving, we get . Hence, and by the Pythagorean Theorem. The total perimeter is . The answer is thus .
Solution 2
In a similar fashion, we encode the angles as complex numbers, so if , then and . So we need only find such that . This will happen when , which simplifies to . Therefore, . By the Pythagorean Theorem, , so the perimeter is , giving us our answer, .
Solution 3
Let , so , , and thus We can then draw the angle bisector of , and let it intersect at Since , Let . Then we see by the Pythagorean Theorem, , , , and By the angle bisector theorem, Substituting in what we know for the lengths of those segments, we see that multiplying by both denominators and squaring both sides yields which simplifies to Substituting this in for x in the equations for and yields and Thus the perimeter is , and the answer is .
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.