Difference between revisions of "1988 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
+ | In [[triangle]] <math>ABC</math>, <math>\tan \angle CAB = 22/7</math>, and the [[altitude]] from <math>A</math> divides <math>BC</math> into [[segment]]s of length 3 and 17. What is the area of triangle <math>ABC</math>? | ||
== Solution == | == Solution == | ||
+ | <center>[[Image:AIME_1988_Solution_07.png]]</center> | ||
+ | |||
+ | Call <math>\angle BAD</math> <math>\alpha</math> and <math>\angle CAD</math> <math>\beta</math>. So, <math>\tan \alpha = \frac {17}{h}</math> and <math>\tan \beta = \frac {3}{h}</math>. Using the tangent addition formula <math>\tan (\alpha + \beta) = \dfrac {\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \beta}</math>, we get <math>\tan (\alpha + \beta) = \dfrac {\frac {20}{h}}{\frac {h^2 - 51}{h^2}} = \frac {22}{7}</math>. | ||
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+ | Simplifying, we get <math>\frac {20h}{h^2 - 51} = \frac {22}{7}</math>. Cross-multiplying and simplifying, we get <math>11h^2-70h-561 = 0</math>. Factoring, we get <math>(11h+51)(h-11) = 0</math>, so we take the positive positive solution, which is <math>h = 11</math>. Therefore, the answer is <math>\frac {20 \cdot 11}{2} = 110</math>, so the answer is <math>\boxed {110}</math>. | ||
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+ | ~Arcticturn | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=1988|num-b=6|num-a=8}} | |
+ | |||
+ | [[Category:Intermediate Trigonometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 22:25, 20 November 2023
Problem
In triangle , , and the altitude from divides into segments of length 3 and 17. What is the area of triangle ?
Solution
Call and . So, and . Using the tangent addition formula , we get .
Simplifying, we get . Cross-multiplying and simplifying, we get . Factoring, we get , so we take the positive positive solution, which is . Therefore, the answer is , so the answer is .
~Arcticturn
See also
1988 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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