Difference between revisions of "2019 AIME II Problems/Problem 1"
Brendanb4321 (talk | contribs) (→Solution) |
Hastapasta (talk | contribs) |
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Line 9: | Line 9: | ||
pair C = (15,8); | pair C = (15,8); | ||
pair D = (-6,8); | pair D = (-6,8); | ||
+ | pair E = (-6,0); | ||
draw(A--B--C--cycle); | draw(A--B--C--cycle); | ||
draw(B--D--A); | draw(B--D--A); | ||
Line 15: | Line 16: | ||
label("$C$",C,dir(60)); | label("$C$",C,dir(60)); | ||
label("$D$",D,dir(120)); | label("$D$",D,dir(120)); | ||
+ | label("$E$",E,dir(-135)); | ||
label("$9$",(A+B)/2,dir(-90)); | label("$9$",(A+B)/2,dir(-90)); | ||
label("$10$",(D+A)/2,dir(-150)); | label("$10$",(D+A)/2,dir(-150)); | ||
Line 21: | Line 23: | ||
label("$17$",(A+C)/2,dir(120)); | label("$17$",(A+C)/2,dir(120)); | ||
− | draw(D-- | + | draw(D--E--A,dotted); |
− | label("$8$",(D+ | + | label("$8$",(D+E)/2,dir(180)); |
− | label("$6$",(A+ | + | label("$6$",(A+E)/2,dir(-90)); |
</asy> | </asy> | ||
- Diagram by Brendanb4321 | - Diagram by Brendanb4321 | ||
Line 29: | Line 31: | ||
Extend <math>AB</math> to form a right triangle with legs <math>6</math> and <math>8</math> such that <math>AD</math> is the hypotenuse and connect the points <math>CD</math> so | Extend <math>AB</math> to form a right triangle with legs <math>6</math> and <math>8</math> such that <math>AD</math> is the hypotenuse and connect the points <math>CD</math> so | ||
− | that you have a rectangle. The base <math>CD</math> of the rectangle will be <math>9+6+6=21</math>. Now, let <math> | + | that you have a rectangle. (We know that <math>\triangle ADE</math> is a <math>6-8-10</math>, since <math>\triangle DEB</math> is an <math>8-15-17</math>.) The base <math>CD</math> of the rectangle will be <math>9+6+6=21</math>. Now, let <math>O</math> be the intersection of <math>BD</math> and <math>AC</math>. This means that <math>\triangle ABO</math> and <math>\triangle DCO</math> are with ratio <math>\frac{21}{9}=\frac73</math>. Set up a proportion, knowing that the two heights add up to 8. We will let <math>y</math> be the height from <math>O</math> to <math>DC</math>, and <math>x</math> be the height of <math>\triangle ABO</math>. |
<cmath>\frac{7}{3}=\frac{y}{x}</cmath> | <cmath>\frac{7}{3}=\frac{y}{x}</cmath> | ||
<cmath>\frac{7}{3}=\frac{8-x}{x}</cmath> | <cmath>\frac{7}{3}=\frac{8-x}{x}</cmath> | ||
Line 100: | Line 102: | ||
- Solution by Duoquinquagintillion | - Solution by Duoquinquagintillion | ||
+ | == Solution 4 == | ||
+ | Let <math>a = \angle{CAB}</math>. By Law of Cosines, | ||
+ | <cmath>\cos a = \frac{17^2+9^2-10^2}{2*9*17} = \frac{15}{17}</cmath> | ||
+ | <cmath>\sin a = \sqrt{1-\cos^2 a} = \frac{8}{17}</cmath> | ||
+ | <cmath>\tan a = \frac{8}{15}</cmath> | ||
+ | <cmath>A = \frac{1}{2}* 9*\frac{9}{2}\tan a = \frac{54}{5}</cmath> | ||
+ | And <math>54+5=\boxed{059}.</math> | ||
+ | |||
+ | - by Mathdummy | ||
+ | |||
+ | == Solution 5 == | ||
+ | Because <math>AD = BC</math> and <math>\angle BAD = \angle ABC</math>, quadrilateral <math>ABCD</math> is cyclic. So, Ptolemy's theorem tells us that | ||
+ | <cmath>AB \cdot CD + BC \cdot AD = AC \cdot BD \implies 9 \cdot CD + 10^2 = 17^2 \implies CD = 21.</cmath> | ||
+ | |||
+ | From here, there are many ways to finish which have been listed above. If we let <math>AB \cap CD = P</math>, then | ||
+ | <cmath>\triangle APB \sim \triangle CPD \implies \frac{AP}{AB} = \frac{CP}{CD} \implies \frac{AP}{9} = \frac{17-AP}{21} \implies AP = 5.1.</cmath> | ||
+ | |||
+ | Using Heron's formula on <math>\triangle ABP</math>, we see that | ||
+ | <cmath>[ABC] = \sqrt{9.6(9.6-5.1)(9.6-5.1)(9.6-9)} = 10.8 = \frac{54}{5}.</cmath> | ||
+ | |||
+ | Thus, our answer is <math>059</math>. ~a.y.711 | ||
+ | |||
+ | == Solution 6 == | ||
+ | |||
+ | Let <math>A=(0,0), B=(9,0)</math>. Now consider <math>C</math>, and if we find the coordinates of <math>C</math>, by symmetry about <math>x=4.5</math>, we can find the coordinates of D. | ||
+ | |||
+ | So let <math>C=(a,b)</math>. So the following equations hold: | ||
+ | |||
+ | <math>\sqrt{(a-9)^2+(b)^2}=17</math>. | ||
+ | |||
+ | <math>\sqrt{a^2+b^2}=10</math>. | ||
+ | |||
+ | Solving by squaring both equations and then subtracting one from the other to eliminate <math>b^2</math>, we get <math>C=(-6,8)</math> because <math>C</math> is in the second quadrant. | ||
+ | |||
+ | Now by symmetry, <math>D=(16, 8)</math>. | ||
+ | |||
+ | So now you can proceed by finding the intersection and then calculating the area directly. We get <math>\boxed{059}</math>. | ||
+ | |||
+ | ~hastapasta | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=II|before=First Problem|num-a=2}} | {{AIME box|year=2019|n=II|before=First Problem|num-a=2}} | ||
+ | [[Category: Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 14:05, 12 January 2023
Contents
Problem
Two different points, and , lie on the same side of line so that and are congruent with , , and . The intersection of these two triangular regions has area , where and are relatively prime positive integers. Find .
Solution
- Diagram by Brendanb4321
Extend to form a right triangle with legs and such that is the hypotenuse and connect the points so
that you have a rectangle. (We know that is a , since is an .) The base of the rectangle will be . Now, let be the intersection of and . This means that and are with ratio . Set up a proportion, knowing that the two heights add up to 8. We will let be the height from to , and be the height of .
This means that the area is . This gets us
-Solution by the Math Wizard, Number Magician of the Second Order, Head of the Council of the Geometers
Solution 2
Using the diagram in Solution 1, let be the intersection of and . We can see that angle is in both and . Since and are congruent by AAS, we can then state and . It follows that and . We can now state that the area of is the area of the area of . Using Heron's formula, we compute the area of . Using the Law of Cosines on angle , we obtain
(For convenience, we're not going to simplify.)
Applying the Law of Cosines on yields This means . Next, apply Heron's formula to get the area of , which equals after simplifying. Subtracting the area of from the area of yields the area of , which is , giving us our answer, which is -Solution by flobszemathguy
Solution 3 (Very quick)
- Diagram by Brendanb4321 extended by Duoquinquagintillion
Begin with the first step of solution 1, seeing is the hypotenuse of a triangle and calling the intersection of and point . Next, notice is the hypotenuse of an triangle. Drop an altitude from with length , so the other leg of the new triangle formed has length . Notice we have formed similar triangles, and we can solve for .
So has area And - Solution by Duoquinquagintillion
Solution 4
Let . By Law of Cosines, And
- by Mathdummy
Solution 5
Because and , quadrilateral is cyclic. So, Ptolemy's theorem tells us that
From here, there are many ways to finish which have been listed above. If we let , then
Using Heron's formula on , we see that
Thus, our answer is . ~a.y.711
Solution 6
Let . Now consider , and if we find the coordinates of , by symmetry about , we can find the coordinates of D.
So let . So the following equations hold:
.
.
Solving by squaring both equations and then subtracting one from the other to eliminate , we get because is in the second quadrant.
Now by symmetry, .
So now you can proceed by finding the intersection and then calculating the area directly. We get .
~hastapasta
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
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.