Difference between revisions of "2008 AMC 12B Problems/Problem 17"
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− | Supposing <math>\angle A=90^\circ</math>, <math>AC</math> is perpendicular to <math>AB</math> and, it follows, to the <math>x</math>-axis, making <math>AC</math> a segment of the line <math>x= | + | Supposing <math>\angle A=90^\circ</math>, <math>AC</math> is perpendicular to <math>AB</math> and, it follows, to the <math>x</math>-axis, making <math>AC</math> a segment of the line <math>x=m</math>. But that would mean that the coordinates of <math>C</math> are <math>(m, m^2)</math>, contradicting the given that points <math>A</math> and <math>C</math> are distinct. So <math>\angle A</math> is not <math>90^\circ</math>. By a similar logic, neither is <math>\angle B</math>. |
This means that <math>\angle C=90^\circ</math> and <math>AC</math> is perpendicular to <math>BC</math>. Let C be the point <math>(n, n^2)</math>. So the slope of <math>BC</math> is the negative reciprocal of the slope of <math>AC</math>, yielding <math>m+n=\frac{1}{m-n}</math> <math>\Rightarrow</math> <math>m^2-n^2=1</math>. | This means that <math>\angle C=90^\circ</math> and <math>AC</math> is perpendicular to <math>BC</math>. Let C be the point <math>(n, n^2)</math>. So the slope of <math>BC</math> is the negative reciprocal of the slope of <math>AC</math>, yielding <math>m+n=\frac{1}{m-n}</math> <math>\Rightarrow</math> <math>m^2-n^2=1</math>. |
Latest revision as of 22:51, 5 October 2023
Problem
Let , and be three distinct points on the graph of such that line is parallel to the -axis and is a right triangle with area . What is the sum of the digits of the -coordinate of ?
Solution
Supposing , is perpendicular to and, it follows, to the -axis, making a segment of the line . But that would mean that the coordinates of are , contradicting the given that points and are distinct. So is not . By a similar logic, neither is .
This means that and is perpendicular to . Let C be the point . So the slope of is the negative reciprocal of the slope of , yielding .
Because is the length of the altitude of triangle from , and is the length of , the area of . Since , . Substituting, , whose digits sum to .
See Also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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