Difference between revisions of "1988 AIME Problems/Problem 7"
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− | 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) = \frac {\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \ beta}</math>, we get <math>\ | + | 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) = \frac {\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \ beta}</math>, we get <math>\dfrac {\frac {20}{h}}{\frac {x^2 - 51}{x^2}}</math>. |
== See also == | == See also == |
Revision as of 23:20, 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
![AIME 1988 Solution 07.png](https://wiki-images.artofproblemsolving.com//d/d6/AIME_1988_Solution_07.png)
Call
and
. So,
and
. Using the tangent addition formula
, we get
.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.