Difference between revisions of "2017 AIME II Problems/Problem 10"
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− | Impose a coordinate system on the diagram where point <math>D</math> is the origin. Therefore <math>A=(0,42)</math>, <math>B=(84,42)</math>, <math>C=(84,0)</math>, and <math>D=(0,0)</math>. Because <math>M</math> is a midpoint and <math>N</math> is a trisection point, <math>M=(0,21)</math> and <math>N=(28,42)</math>. The equation for line <math>DN</math> is <math>y=\frac{3}{2}x</math> and the equation for line <math>CM</math> is <math>\frac{1}{84}x+\frac{1}{21}y=1</math>, so their intersection, point <math>O</math>, is <math>(12,18)</math>. Using the shoelace formula on quadrilateral <math>BCON</math>, or or drawing diagonal <math>\overline{BO}</math> and using <math>\frac 12 bh</math>, we find that its area is <math>2184</math>. Therefore the area of triangle <math>BCP</math> is <math>\frac{2184}{2}</math> and the distance from <math>P</math> to line <math>BC</math> is <math>52</math> and its <math>x</math>-coordinate is <math>32</math>. Because <math>P</math> lies on the equation <math>\frac{1}{84}x+\frac{1}{21}y=1</math>, <math>P</math>'s <math>y</math>-coordinate is <math>13</math>, which is also the height of <math>CDP</math>. Therefore the area of <math>CDP</math> is <math>\frac{1}{2} \cdot 13 \cdot 84=\boxed{546}</math>. | + | Impose a coordinate system on the diagram where point <math>D</math> is the origin. Therefore <math>A=(0,42)</math>, <math>B=(84,42)</math>, <math>C=(84,0)</math>, and <math>D=(0,0)</math>. Because <math>M</math> is a midpoint and <math>N</math> is a trisection point, <math>M=(0,21)</math> and <math>N=(28,42)</math>. The equation for line <math>DN</math> is <math>y=\frac{3}{2}x</math> and the equation for line <math>CM</math> is <math>\frac{1}{84}x+\frac{1}{21}y=1</math>, so their intersection, point <math>O</math>, is <math>(12,18)</math>. Using the shoelace formula on quadrilateral <math>BCON</math>, or or drawing diagonal <math>\overline{BO}</math> and using <math>\frac 12 bh</math>, we find that its area is <math>2184</math>. Therefore the area of triangle <math>BCP</math> is <math>\frac{2184}{2}</math> =1092. Using A=<math>\frac 12 bh</math>, we get 1092= = 42*h. Simplifying, we get h=52. This means that the x-coordinate of P= 84-52=32. Since P lies on <math>\frac{1}{84}x+\frac{1}{21}y=1</math>, you can solve and get that the y-coordinate of P is 13. Therefore the area of <math>CDP</math> is <math>\frac{1}{2} \cdot 13 \cdot 84=\boxed{546}</math>. |
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+ | Solution Altered By conantwiz2023 | ||
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+ | and the distance from <math>P</math> to line <math>BC</math> is <math>52</math> and its <math>x</math>-coordinate is <math>32</math>. Because <math>P</math> lies on the equation <math>\frac{1}{84}x+\frac{1}{21}y=1</math>, <math>P</math>'s <math>y</math>-coordinate is <math>13</math>, which is also the height of <math>CDP</math>. Therefore the area of <math>CDP</math> is <math>\frac{1}{2} \cdot 13 \cdot 84=\boxed{546}</math>. | ||
=See Also= | =See Also= | ||
{{AIME box|year=2017|n=II|num-b=9|num-a=11}} | {{AIME box|year=2017|n=II|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:47, 5 March 2018
Problem
Rectangle has side lengths and . Point is the midpoint of , point is the trisection point of closer to , and point is the intersection of and . Point lies on the quadrilateral , and bisects the area of . Find the area of .
Solution
Impose a coordinate system on the diagram where point is the origin. Therefore , , , and . Because is a midpoint and is a trisection point, and . The equation for line is and the equation for line is , so their intersection, point , is . Using the shoelace formula on quadrilateral , or or drawing diagonal and using , we find that its area is . Therefore the area of triangle is =1092. Using A=, we get 1092= = 42*h. Simplifying, we get h=52. This means that the x-coordinate of P= 84-52=32. Since P lies on , you can solve and get that the y-coordinate of P is 13. Therefore the area of is .
Solution Altered By conantwiz2023
and the distance from to line is and its -coordinate is . Because lies on the equation , 's -coordinate is , which is also the height of . Therefore the area of is .
See Also
2017 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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.