Difference between revisions of "2013 AIME II Problems/Problem 13"
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Area of the triangle is ah = <math>3\sqrt{7}</math>, giving us an answer of <math>\boxed{010}</math>. | Area of the triangle is ah = <math>3\sqrt{7}</math>, giving us an answer of <math>\boxed{010}</math>. | ||
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Let <math>BD = x</math>. Then <math>CD = 3x</math> and <math>AC = 4x</math>. Also, let <math>AE = ED = l</math>. Using Stewart's Theorem on <math>\bigtriangleup CEB</math> gives us the equation <math>(x)(3x)(4x) + (4x)(l^2) = 27x + 7x</math> or, after simplifying, <math>4l^2 = 34 - 12x^2</math>. We use Stewart's again on <math>\bigtriangleup CAD</math>: <math>(l)(l)(2l) + 7(2l) = (16x^2)(l) + (9x^2)(l)</math>, which becomes <math>2l^2 = 25x^2 - 14</math>. Substituting <math>2l^2 = 17 - 6x^2</math>, we see that <math>31x^2 = 31</math>, or <math>x = 1</math>. Then <math>l^2 = \frac{11}{2}</math>. | Let <math>BD = x</math>. Then <math>CD = 3x</math> and <math>AC = 4x</math>. Also, let <math>AE = ED = l</math>. Using Stewart's Theorem on <math>\bigtriangleup CEB</math> gives us the equation <math>(x)(3x)(4x) + (4x)(l^2) = 27x + 7x</math> or, after simplifying, <math>4l^2 = 34 - 12x^2</math>. We use Stewart's again on <math>\bigtriangleup CAD</math>: <math>(l)(l)(2l) + 7(2l) = (16x^2)(l) + (9x^2)(l)</math>, which becomes <math>2l^2 = 25x^2 - 14</math>. Substituting <math>2l^2 = 17 - 6x^2</math>, we see that <math>31x^2 = 31</math>, or <math>x = 1</math>. Then <math>l^2 = \frac{11}{2}</math>. |
Revision as of 12:42, 8 August 2021
Contents
[hide]Problem 13
In , , and point is on so that . Let be the midpoint of . Given that and , the area of can be expressed in the form , where and are positive integers and is not divisible by the square of any prime. Find .
Solutions
Solution 1
We can set . Set , therefore . Thereafter, by Stewart's Theorem on and cevian , we get . Also apply Stewart's Theorem on with cevian . After simplification, . Therefore, . Finally, note that (using [] for area) , because of base-ratios. Using Heron's Formula on , as it is simplest, we see that , so your answer is .
Solution 2
After drawing the figure, we suppose , so that , , and .
Using Law of Cosines for and ,we get
So, , we get
Using Law of Cosines in , we get
So,
Using Law of Cosines in and , we get
, and according to , we can get
Using and , we can solve and .
Finally, we use Law of Cosines for ,
then , so the height of this is .
Then the area of is , so the answer is .
Solution 3
Let be the foot of the altitude from with other points labelled as shown below. Now we proceed using mass points. To balance along the segment , we assign a mass of and a mass of . Therefore, has a mass of . As is the midpoint of , we must assign a mass of as well. This gives a mass of and a mass of .
Now let be the base of the triangle, and let be the height. Then as , and as , we know that Also, as , we know that . Therefore, by the Pythagorean Theorem on , we know that
Also, as , we know that . Furthermore, as , and as , we know that and , so . Therefore, by the Pythagorean Theorem on , we get Solving this system of equations yields and . Therefore, the area of the triangle is , giving us an answer of .
Solution 4
Let the coordinates of A, B and C be (-a, 0), (a, 0) and (0, h) respectively. Then and implies ; implies Solve this system of equations simultaneously, and . Area of the triangle is ah = , giving us an answer of .
Solution 5
Let . Then and . Also, let . Using Stewart's Theorem on gives us the equation or, after simplifying, . We use Stewart's again on : , which becomes . Substituting , we see that , or . Then .
We now use Law of Cosines on . . Plugging in for and , , so . Using the Pythagorean trig identity , , so .
, and our answer is .
Note to writter: Couldn't we just use Heron's formula for after is solved then noticing that ?
Solution 5 (Barycentric Coordinates)
Let ABC be the reference triangle, with , , and . We can easily calculate and subsequently . Using distance formula on and gives
But we know that , so we can substitute and now we have two equations and two variables. So we can clear the denominators and prepare to cancel a variable:
Then we add the equations to get
Then plugging gives and . Then the height from is , and the area is and our answer is .
See Also
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.