Difference between revisions of "2007 AIME I Problems/Problem 12"

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== Problem ==
 
== Problem ==
In [[isosceles triangle]] <math>\triangle ABC</math>, <math>A</math> is located at the [[origin]] and <math>B</math> is located at (20,0).  Point <math>C</math> is in the [[first quadrant]] with <math>\displaystyle AC = BC</math> and angle <math>BAC = 75^{\circ}</math>.  If triangle <math>ABC</math> is rotated counterclockwise about point <math>A</math> until the image of <math>C</math> lies on the positive <math>y</math>-axis, the area of the region common to the original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>\displaystyle p,q,r,s</math> are integers.  Find <math>\frac{p-q+r-s}2</math>.
+
In [[isosceles triangle]] <math>\triangle ABC</math>, <math>A</math> is located at the [[origin]] and <math>B</math> is located at <math>(20,0)</math>.  Point <math>C</math> is in the first quadrant with <math>AC = BC</math> and angle <math>BAC = 75^{\circ}</math>.  If triangle <math>ABC</math> is rotated counterclockwise about point <math>A</math> until the image of <math>C</math> lies on the positive <math>y</math>-axis, the area of the region common to the original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>p,q,r,s</math> are integers.  Find <math>\frac{p-q+r-s}2</math>.
 +
 
 +
 
 +
 
 +
== Solution ==
  
 
{|
 
{|
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| [[Image:AIME I 2007-12.png]]
|__TOC__
 
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 
|[[Image:AIME I 2007-12.png]]
 
 
|}
 
|}
== Solution ==
+
 
 
=== Solution 1 ===
 
=== Solution 1 ===
Call the [[vertex|vertices]] of the new triangle <math>\displaystyle AB'C'</math> (<math>A</math>, the origin, is a vertex of both triangles). <math>\displaystyle B'C'</math> and <math>\displaystyle AB</math> intersect at a single point, <math>D</math>. <math>\displaystyle BC</math> intersect at two points; the one with the higher y-coordinate will be <math>E</math>, and the other <math>F</math>. The intersection of the two triangles is a [[quadrilateral]] <math>\displaystyle ADEF</math>. Notice that we can find this area by subtracting <math>[\triangle ADB'] - [\triangle EFB']</math>.  
+
Let the new triangle be <math>\triangle AB'C'</math> (<math>A</math>, the origin, is a vertex of both triangles). Let <math>\overline{B'C'}</math> intersect with <math>\overline{AC}</math> at point <math>D</math>, <math>\overline{BC}</math> intersect with <math>\overline{B'C'}</math> at <math>E</math>, and <math>\overline{BC}</math> intersect with <math>\overline{AB'}</math> at <math>F</math>. The region common to both triangles is the [[quadrilateral]] <math>ADEF</math>. Notice that <math>[ADEF] = [\triangle ADB'] - [\triangle EFB']</math>, where we let <math>[\ldots]</math> denote area.  
 +
 
 +
<center>To find <math>[\triangle ADB']</math>:</center>
  
Since <math>\displaystyle \angle B'AC'</math> and <math>\displaystyle \angle BAC</math> both have measures <math>75^{\circ}</math>, both of their [[complement]]s are <math>15^{\circ}</math>, and <math>\angle DAC' = 90 - 2(15) = 60^{\circ}</math>. We know that <math>C'B'A = 75^{\circ}</math>, and since the angles of a triangle add up to <math>180^{\circ}</math>, we find that <math>ADB' = 180 - 60 - 75 = 45^{\circ}</math>.
+
Since <math>\angle B'AC'</math> and <math>\angle BAC</math> both have measures <math>75^{\circ}</math>, both of their [[complement]]s are <math>15^{\circ}</math>, and <math>\angle DAB' = 90 - 2(15) = 60^{\circ}</math>. We know that <math>\angle DB'A = 75^{\circ}</math>, so <math>\angle ADB' = 180 - 60 - 75 = 45^{\circ}</math>.
  
So <math>\displaystyle ADB'</math> is a <math>45 - 60 - 75 \triangle</math>. It can be solved by drawing an [[altitude]] splitting the <math>75^{\circ}</math> angle into <math>30^{\circ}</math> and <math>45^{\circ}</math> angles &ndash; this forms a <math>\displaystyle 30-60-90</math> [[right triangle]] and a <math>\displaystyle 45-45-90</math> isosceles right triangle. Since we know that <math>\displaystyle DB' = 20</math>, the base of the <math>\displaystyle 30-60-90</math> triangle is <math>10</math>, the height is <math>10\sqrt{3}</math>, and the base of the <math>\displaystyle 45-45-90</math> is <math>10\sqrt{3}</math>. Thus, the total area of <math>[\triangle ADB'] = \frac{1}{2}(10\sqrt{3})(10\sqrt{3} + 10) = 150 + 50\sqrt{3}</math>.
+
Thus <math>\triangle ADB'</math> is a <math>45 - 60 - 75 \triangle</math>. It can be solved by drawing an [[altitude]] splitting the <math>75^{\circ}</math> angle into <math>30^{\circ}</math> and <math>45^{\circ}</math> angles, forming a <math>30-60-90</math> [[right triangle]] and a <math>45-45-90</math> isosceles right triangle. Since we know that <math>AB' = 20</math>, the base of the <math>30-60-90</math> triangle is <math>10</math>, the base of the <math>45-45-90</math> is <math>10\sqrt{3}</math>, and their common height is <math>10\sqrt{3}</math>. Thus, the total area of <math>[\triangle ADB'] = \frac{1}{2}(10\sqrt{3})(10\sqrt{3} + 10) = \boxed{150 + 50\sqrt{3}}</math>.
  
Now, we need to find <math>[\triangle EFB']</math>, which is a <math>\displaystyle 15-75-90</math> right triangle. We can find its base by subtracting <math>AF</math> from <math>20</math>. <math>\triangle AFB</math> is also a <math>\displaystyle 15-75-90</math> triangle, so we find that <math>AF = 20\sin 75 = 20 \sin (30 + 45) = 20\frac{\sqrt{2} + \sqrt{6}}4 = 5\sqrt{2} + 5\sqrt{6}</math>. <math>FB' = 20 - AF = 20 - 5\sqrt{2} - 5\sqrt{6}</math>.
+
<center>To find <math>[\triangle EFB']</math>:</center>
  
To solve <math>[\triangle EFB']</math>, note that <math>\displaystyle [\triangle EFB'] = \frac{1}{2} FB' \cdot EF = \frac{1}{2} FB' \cdot (\tan 75 FB')</math>. Through [[algebra]], we can calculate <math>(FB')^2 \cdot \tan 75</math>:
+
Since <math>\triangle AFB</math> is also a <math>15-75-90</math> triangle,
:<math>\displaystyle \frac{1}{2}\tan (30 + 45) \cdot (20 - 5\sqrt{2} - 5\sqrt{6})^2</math>
+
<center><math>AF = 20\sin 75 = 20 \sin (30 + 45) = 20\left(\frac{\sqrt{2} + \sqrt{6}}4\right) = 5\sqrt{2} + 5\sqrt{6}</math></center>
:<math>\displaystyle = \frac{1}{2} \left(\frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}}}\right) \left[400 - 100\sqrt{2} - 100\sqrt{6} - 100\sqrt{2} + 50 + 50\sqrt{3} - 100\sqrt{6} + 50\sqrt{3} + 150\right]</math>
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and
:<math>= \frac{1}{2}(2 + \sqrt{3})[600 - 200\sqrt{2} - 200\sqrt{6} + 100\sqrt{3}]</math>
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<center><math>FB' = AB' - AF = 20 - 5\sqrt{2} - 5\sqrt{6}</math></center>
:<math>=- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750</math>
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Since <math>[\triangle EFB'] = \frac{1}{2} (FB' \cdot EF) = \frac{1}{2} (FB') (FB' \tan 75^{\circ})</math>. With some horrendous [[algebra]], we can calculate
 +
<cmath>\begin{align*}
 +
[\triangle EFB'] &= \frac{1}{2}\tan (30 + 45) \cdot (20 - 5\sqrt{2} - 5\sqrt{6})^2 \\
 +
&= 25 \left(\frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}}}\right) \left(8 - 2\sqrt{2} - 2\sqrt{6} - 2\sqrt{2} + 1 + \sqrt{3} - 2\sqrt{6} + \sqrt{3} + 3\right) \\
 +
&= 25(2 + \sqrt{3})(12 - 4\sqrt{2} - 4\sqrt{6} + 2\sqrt{3}) \\
 +
[\triangle EFB'] &= \boxed{- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750}.
 +
\end{align*}</cmath>
  
To finish, find <math>[ADEF] = [\triangle ADB'] - [\triangle EFB']</math><math>= \left(150 + 50\sqrt{3}\right) - \left(-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} + 750\right)</math><math>=500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>. The solution is <math>\frac{500 + 350 + 300 + 600}2 = \frac{1750}2 = 875</math>.
+
To finish,
 +
<cmath>\begin{align*}
 +
[ADEF] &= [\triangle ADB'] - [\triangle EFB']\\
 +
      &= \left(150 + 50\sqrt{3}\right) - \left(-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} + 750\right)\\
 +
      &=500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600\\ \end{align*}</cmath>
 +
Hence, <math>\frac{p-q+r-s}{2} = \frac{500 + 350 + 300 + 600}2 = \frac{1750}2 = \boxed{\boxed{875}}</math>.
  
 
=== Solution 2 ===
 
=== Solution 2 ===
Redefine the points in the same manner as the last time (<math>\displaystyle \triangle AB'C'</math>, intersect at <math>D</math>, <math>E</math>, and <math>F</math>). This time, notice that <math>[ADEF] = [\triangle AB'C'] - ([\triangle ADC'] + [\triangle EFB']</math>.
+
Redefine the points in the same manner as the last time (<math>\triangle AB'C'</math>, intersect at <math>D</math>, <math>E</math>, and <math>F</math>). This time, notice that <math>[ADEF] = [\triangle AB'C'] - ([\triangle ADC'] + [\triangle EFB']</math>.
  
The area of <math>[\triangle AB'C'] = [\triangle ABC]</math>. The [[altitude]] of <math>\triangle ABC</math> is clearly <math>10 \tan 75 = 10 \tan (30 + 45)</math>. The [[trigonometric identity|tangent addition rule]] yields <math>10(2 + \sqrt{3})</math> (see above). Thus, <math>\displaystyle [\triangle ABC] = \frac{1}{2} 20 \cdot (20 + 10\sqrt{3}) = 200 + 100\sqrt{3}</math>.  
+
The area of <math>[\triangle AB'C'] = [\triangle ABC]</math>. The [[altitude]] of <math>\triangle ABC</math> is clearly <math>10 \tan 75 = 10 \tan (30 + 45)</math>. The [[trigonometric identity|tangent addition rule]] yields <math>10(2 + \sqrt{3})</math> (see above). Thus, <math>[\triangle ABC] = \frac{1}{2} 20 \cdot (20 + 10\sqrt{3}) = 200 + 100\sqrt{3}</math>.  
  
The area of <math>[\triangle ADC']</math> (with a side on the y-axis) can be found by splitting it into two triangles, <math>30-60-90</math> and <math>15-75-90</math> [[right triangle]]s. <math>AC' = AC = \frac{10}{\sin 75}</math>. The [[trigonometric identity|sine subtraction rule]] shows that <math>\frac{10}{\sin 15} = \frac{10}{\frac{\sqrt{6} - \sqrt{2}}4} = \frac{40}{\sqrt{6} - \sqrt{2}} = 10(\sqrt{6} + \sqrt{2})</math>. <math>AC'</math>, in terms of the height of <math>\triangle ADC'</math>, is equal to <math>h(\sqrt{3} + \tan 75) = h(\sqrt{3} + 2 + \sqrt{3})</math>. <math>[ADC'] = \frac 12 AC' \cdot h = \frac 12 (10\sqrt{6} + 10\sqrt{2})(\frac{10\sqrt{6} + 10\sqrt{2}}{2\sqrt{3} + 2}</math><math>= \frac{(800 + 400\sqrt{3})}{(2 + \sqrt{3})}\cdot\frac{2 - \sqrt{3}}{2-\sqrt{3}}</math><math>= \frac{400\sqrt{3} + 400}8 = 50\sqrt{3} + 50</math>.
+
The area of <math>[\triangle ADC']</math> (with a side on the y-axis) can be found by splitting it into two triangles, <math>30-60-90</math> and <math>15-75-90</math> [[right triangle]]s. <math>AC' = AC = \frac{10}{\sin 15}</math>. The [[trigonometric identity|sine subtraction rule]] shows that <math>\frac{10}{\sin 15} = \frac{10}{\frac{\sqrt{6} - \sqrt{2}}4} = \frac{40}{\sqrt{6} - \sqrt{2}} = 10(\sqrt{6} + \sqrt{2})</math>. <math>AC'</math>, in terms of the height of <math>\triangle ADC'</math>, is equal to <math>h(\sqrt{3} + \tan 75) = h(\sqrt{3} + 2 + \sqrt{3})</math>.  
 +
<cmath>\begin{align*}
 +
[ADC'] &= \frac 12 AC' \cdot h \\
 +
&= \frac 12 (10\sqrt{6} + 10\sqrt{2})\left(\frac{10\sqrt{6} + 10\sqrt{2}}{2\sqrt{3} + 2}\right) \\
 +
&= \frac{(800 + 400\sqrt{3})}{(2 + \sqrt{3})}\cdot\frac{2 - \sqrt{3}}{2-\sqrt{3}} \\
 +
&= \frac{400\sqrt{3} + 400}8 = 50\sqrt{3} + 50
 +
\end{align*}</cmath>
  
 
The area of <math>[\triangle EFB']</math> was found in the previous solution to be <math>- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750</math>.  
 
The area of <math>[\triangle EFB']</math> was found in the previous solution to be <math>- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750</math>.  
  
Therefore, <math>[ADEF] = (200 + 100\sqrt{3}) - \left((50 + 50\sqrt{3}) + (-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750)\right) = 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>, and our answer is <math>875</math>.
+
Therefore, <math>[ADEF] </math> <math>= (200 + 100\sqrt{3}) - \left((50 + 50\sqrt{3}) + (-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750)\right)</math> <math>= 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>, and our answer is <math>\boxed{875}</math>.
  
 
=== Solution 3 ===
 
=== Solution 3 ===
 
[[Image:AIME I 2007-12b.png|left]]  
 
[[Image:AIME I 2007-12b.png|left]]  
  
Call the points of the intersections of the triangles as <math>D</math>, <math>E</math>, and <math>F</math> as noted in the above diagram (the points are different from those in the diagram for solution 1). <math>\overline{AD}</math> [[bisect]]s <math>\displaystyle \angle EDE'</math>. Through [[HL]] congruency, we can find that <math>\triangle AED</math> is [[congruent]] to <math>\triangle AE'D</math>. This divides the region <math>AEDF</math> (which we are trying to solve for) into two congruent triangles and an [[isosceles triangle|isosceles]] [[right triangle]].   
+
Call the points of the intersections of the triangles <math>D</math>, <math>E</math>, and <math>F</math> as noted in the diagram (the points are different from those in the diagram for solution 1). <math>\overline{AD}</math> [[bisect]]s <math>\angle EDE'</math>.
 +
 
 +
Through [[HL]] congruency, we can find that <math>\triangle AED</math> is [[congruent]] to <math>\triangle AE'D</math>. This divides the region <math>AEDF</math> (which we are trying to solve for) into two congruent triangles and an [[isosceles triangle|isosceles]] [[right triangle]].   
 +
<center><math>AE = 20 \cos 15 = 20 \cos (45 - 30) = 20 \cdot \frac{\sqrt{6} + \sqrt{2}}{4} = 5\sqrt{6} + 5\sqrt{2}</math></center>
 +
Since <math>FE' = AE' = AE</math>, we find that <math>[AE'F] = \frac 12 (5\sqrt{6} + 5\sqrt{2})^2 = 100 + 50\sqrt{3}</math>.
  
<math>AE = 20 \cos 15 = 20 \cos (45 - 30) = 20 \cdot \frac{\sqrt{6} + \sqrt{2}}{4} = 5\sqrt{6} + 5\sqrt{2}</math>. Since <math>FE' = AE' = AE</math>, we find that <math>[AE'F] = \frac 12 (5\sqrt{6} + 5\sqrt{2})^2 = 100 + 50\sqrt{3}</math>.  
+
Now, we need to find <math>[AED] = [AE'D]</math>. The acute angles of the triangles are <math>\frac{15}{2}</math> and <math>90 - \frac{15}{2}</math>. By repeated application of the [[trigonometric identity|half-angle formula]], we can find that <math>\tan \frac{15}{2} = \sqrt{2} - \sqrt{3} + \sqrt{6} - 2</math>.
  
Now, we need to find <math>\displaystyle [AED] = [AE'D]</math>. The acute angles of the triangles are <math>\frac{15}{2},\ 90 - \frac{15}{2}</math>. By repeated application of the [[trigonometric identity|half-angle formula]], we can find that <math>\tan \frac{15}{2} = \sqrt{2} - \sqrt{3} + \sqrt{6} - 2</math>. The area of <math>[AED] = \frac 12 \left(20 \cos 15\right)^2 \left(\tan \frac{15}{2}\right)</math> Thus, <math>[AED] + [AE'D] = 2\left(\frac 12((5\sqrt{6} + 5\sqrt{2})^2 \cdot (\sqrt{2} - \sqrt{3} + \sqrt{6} - 2))\right)</math>. This eventually simplifies to <math>500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>.  
+
The area of <math>[AED] = \frac 12 \left(20 \cos 15\right)^2 \left(\tan \frac{15}{2}\right)</math>. Thus, <math>[AED] + [AE'D] = 2\left(\frac 12((5\sqrt{6} + 5\sqrt{2})^2 \cdot (\sqrt{2} - \sqrt{3} + \sqrt{6} - 2))\right)</math>, which eventually simplifies to <math>500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>.  
  
Adding them together, we find that the solution is <math>[AEDF] = [AE'F] + [AED] + [AE'D] = 100 + 50\sqrt{3} + 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600 = 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>.
+
Adding them together, we find that the solution is <math>[AEDF] = [AE'F] + [AED] + [AE'D]</math> <math>= 100 + 50\sqrt{3} + 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600=</math> <math>= 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600</math>, and the answer is <math>\boxed{875}</math>.
  
 
=== Solution 4 ===
 
=== Solution 4 ===
 
From the given information, calculate the coordinates of all the points (by finding the equations of the lines, equating). Then, use the [[shoelace]] method to calculate the area of the intersection.
 
From the given information, calculate the coordinates of all the points (by finding the equations of the lines, equating). Then, use the [[shoelace]] method to calculate the area of the intersection.
  
*<math>\overline{AC}</math>: <math>y = \sqrt{3}x</math>
+
*<math>\overline{AC}</math>: <math>y = (\tan 75) x = (2 + \sqrt{3})x</math>
*<math>\overline{AB'}</math>: <math>y = (\tan 15) x = \frac{\sqrt{6} - \sqrt{2}}{4}x</math>
+
*<math>\overline{AB'}</math>: <math>y = (\tan 15) x = (2 - \sqrt{3})x</math>
*<math>\overline{BC}</math>: It passes thru <math>(20,0)</math>, and has a slope of <math>-\tan75 = -\frac{\sqrt{6} + \sqrt{2}}{4}</math>. The equation of its line is  <math>y = -\frac{\sqrt{6} + \sqrt{2}}{4}x + 5\sqrt{6} + 5\sqrt{2}</math>.  
+
*<math>\overline{BC}</math>: It passes thru <math>(20,0)</math>, and has a slope of <math>-\tan75 = -(2 + \sqrt{3})</math>. The equation of its line is  <math>y = (2+\sqrt{3})(20 - x)</math>.  
*<math>\overline{B'C'}</math>: <math>AC' = AC = \frac{10}{\sin 75} = 10\sqrt{6} + 10\sqrt{2}</math>, so it passes thru point <math>(0, 10\sqrt{6} + 10\sqrt{2})</math>. It has a slope of <math>-\tan 60 = -\sqrt{3}</math>. So the equation of its line is <math>y = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>.
+
*<math>\overline{B'C'}</math>: <math>AC' = AC = \frac{10}{\cos 75} = 10\sqrt{6} + 10\sqrt{2}</math>, so it passes thru point <math>(0, 10\sqrt{6} + 10\sqrt{2})</math>. It has a slope of <math>-\tan 60 = -\sqrt{3}</math>. So the equation of its line is <math>y = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>.
  
 
Now, we can equate the equations to find the intersections of all the points.
 
Now, we can equate the equations to find the intersections of all the points.
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*<math>A (0,0)</math>
 
*<math>A (0,0)</math>
  
*<math>D</math> is the intersection of <math>\overline{AB'},\ \overline{B'C'}</math>. <math>\frac{\sqrt{6} - \sqrt{2}}{4}x = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>. Therefore, <math>x = \frac{40\sqrt{6} + 40\sqrt{2}}{\sqrt{6} - \sqrt{2} + 4\sqrt{3}}</math>.
+
*<math>D</math> is the intersection of <math>\overline{BC},\ \overline{B'C'}</math>. <math>(2+\sqrt{3})(20-x) = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>. Therefore, <math>x = 5(4 + 2\sqrt{3}-\sqrt{6}-\sqrt{2})</math>, <math>y = 5(3\sqrt{6} + 5\sqrt{2}-4\sqrt{3}-6)</math>.
 +
 
 +
*<math>E</math> is the intersection of <math>\overline{AB'},\ \overline{BC}</math>. <math>(2 - \sqrt{3})x =(2+\sqrt{3})(20-x) </math>. Therefore, <math>x = 5(2+\sqrt{3})</math>, <math>y = 5</math>.
  
*<math>E</math> is the intersection of <math>\overline{AB'},\ \overline{BC}</math>. <math>\frac{\sqrt{6} - \sqrt{2}}{4}x = -\frac{\sqrt{6} + \sqrt{2}}{4}x + 5\sqrt{6} + 5\sqrt{2}</math>. Therefore, <math>x = 10 + \frac{10\sqrt{3}}{3}</math>.
+
*<math>F</math> is the intersection of <math>\overline{AC},\ \overline{B'C'}</math>. <math>(2+\sqrt{3})x = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>. Therefore, <math>x = 5\sqrt{2}</math>, <math>y = 10\sqrt{2}+ 5\sqrt{6}</math>.
  
*<math>F</math> is the intersection of <math>\overline{AC},\ \overline{B'C'}</math>. <math>\sqrt{3}x = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})</math>. Therefore, <math>x = \frac{15\sqrt{2} + 5\sqrt{6}}{3}</math>, and <math>y = 5\sqrt{6} + 5\sqrt{2}</math>.
+
We take these points and tie them together by shoelace, and the answer should come out to be <math>\boxed{875}</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 19:59, 19 February 2019

Problem

In isosceles triangle $\triangle ABC$, $A$ is located at the origin and $B$ is located at $(20,0)$. Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\circ}$. If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s$, where $p,q,r,s$ are integers. Find $\frac{p-q+r-s}2$.


Solution

AIME I 2007-12.png

Solution 1

Let the new triangle be $\triangle AB'C'$ ($A$, the origin, is a vertex of both triangles). Let $\overline{B'C'}$ intersect with $\overline{AC}$ at point $D$, $\overline{BC}$ intersect with $\overline{B'C'}$ at $E$, and $\overline{BC}$ intersect with $\overline{AB'}$ at $F$. The region common to both triangles is the quadrilateral $ADEF$. Notice that $[ADEF] = [\triangle ADB'] - [\triangle EFB']$, where we let $[\ldots]$ denote area.

To find $[\triangle ADB']$:

Since $\angle B'AC'$ and $\angle BAC$ both have measures $75^{\circ}$, both of their complements are $15^{\circ}$, and $\angle DAB' = 90 - 2(15) = 60^{\circ}$. We know that $\angle DB'A = 75^{\circ}$, so $\angle ADB' = 180 - 60 - 75 = 45^{\circ}$.

Thus $\triangle ADB'$ is a $45 - 60 - 75 \triangle$. It can be solved by drawing an altitude splitting the $75^{\circ}$ angle into $30^{\circ}$ and $45^{\circ}$ angles, forming a $30-60-90$ right triangle and a $45-45-90$ isosceles right triangle. Since we know that $AB' = 20$, the base of the $30-60-90$ triangle is $10$, the base of the $45-45-90$ is $10\sqrt{3}$, and their common height is $10\sqrt{3}$. Thus, the total area of $[\triangle ADB'] = \frac{1}{2}(10\sqrt{3})(10\sqrt{3} + 10) = \boxed{150 + 50\sqrt{3}}$.

To find $[\triangle EFB']$:

Since $\triangle AFB$ is also a $15-75-90$ triangle,

$AF = 20\sin 75 = 20 \sin (30 + 45) = 20\left(\frac{\sqrt{2} + \sqrt{6}}4\right) = 5\sqrt{2} + 5\sqrt{6}$

and

$FB' = AB' - AF = 20 - 5\sqrt{2} - 5\sqrt{6}$

Since $[\triangle EFB'] = \frac{1}{2} (FB' \cdot EF) = \frac{1}{2} (FB') (FB' \tan 75^{\circ})$. With some horrendous algebra, we can calculate \begin{align*} [\triangle EFB'] &= \frac{1}{2}\tan (30 + 45) \cdot (20 - 5\sqrt{2} - 5\sqrt{6})^2 \\ &= 25 \left(\frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}}}\right) \left(8 - 2\sqrt{2} - 2\sqrt{6} - 2\sqrt{2} + 1 + \sqrt{3} - 2\sqrt{6} + \sqrt{3} + 3\right) \\ &= 25(2 + \sqrt{3})(12 - 4\sqrt{2} - 4\sqrt{6} + 2\sqrt{3}) \\ [\triangle EFB'] &= \boxed{- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750}. \end{align*}

To finish, \begin{align*} [ADEF] &= [\triangle ADB'] - [\triangle EFB']\\        &= \left(150 + 50\sqrt{3}\right) - \left(-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} + 750\right)\\        &=500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600\\ \end{align*} Hence, $\frac{p-q+r-s}{2} = \frac{500 + 350 + 300 + 600}2 = \frac{1750}2 = \boxed{\boxed{875}}$.

Solution 2

Redefine the points in the same manner as the last time ($\triangle AB'C'$, intersect at $D$, $E$, and $F$). This time, notice that $[ADEF] = [\triangle AB'C'] - ([\triangle ADC'] + [\triangle EFB']$.

The area of $[\triangle AB'C'] = [\triangle ABC]$. The altitude of $\triangle ABC$ is clearly $10 \tan 75 = 10 \tan (30 + 45)$. The tangent addition rule yields $10(2 + \sqrt{3})$ (see above). Thus, $[\triangle ABC] = \frac{1}{2} 20 \cdot (20 + 10\sqrt{3}) = 200 + 100\sqrt{3}$.

The area of $[\triangle ADC']$ (with a side on the y-axis) can be found by splitting it into two triangles, $30-60-90$ and $15-75-90$ right triangles. $AC' = AC = \frac{10}{\sin 15}$. The sine subtraction rule shows that $\frac{10}{\sin 15} = \frac{10}{\frac{\sqrt{6} - \sqrt{2}}4} = \frac{40}{\sqrt{6} - \sqrt{2}} = 10(\sqrt{6} + \sqrt{2})$. $AC'$, in terms of the height of $\triangle ADC'$, is equal to $h(\sqrt{3} + \tan 75) = h(\sqrt{3} + 2 + \sqrt{3})$. \begin{align*} [ADC'] &= \frac 12 AC' \cdot h \\ &= \frac 12 (10\sqrt{6} + 10\sqrt{2})\left(\frac{10\sqrt{6} + 10\sqrt{2}}{2\sqrt{3} + 2}\right) \\ &= \frac{(800 + 400\sqrt{3})}{(2 + \sqrt{3})}\cdot\frac{2 - \sqrt{3}}{2-\sqrt{3}} \\ &= \frac{400\sqrt{3} + 400}8 = 50\sqrt{3} + 50 \end{align*}

The area of $[\triangle EFB']$ was found in the previous solution to be $- 500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750$.

Therefore, $[ADEF]$ $= (200 + 100\sqrt{3}) - \left((50 + 50\sqrt{3}) + (-500\sqrt{2} + 400\sqrt{3} - 300\sqrt{6} +750)\right)$ $= 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600$, and our answer is $\boxed{875}$.

Solution 3

AIME I 2007-12b.png

Call the points of the intersections of the triangles $D$, $E$, and $F$ as noted in the diagram (the points are different from those in the diagram for solution 1). $\overline{AD}$ bisects $\angle EDE'$.

Through HL congruency, we can find that $\triangle AED$ is congruent to $\triangle AE'D$. This divides the region $AEDF$ (which we are trying to solve for) into two congruent triangles and an isosceles right triangle.

$AE = 20 \cos 15 = 20 \cos (45 - 30) = 20 \cdot \frac{\sqrt{6} + \sqrt{2}}{4} = 5\sqrt{6} + 5\sqrt{2}$

Since $FE' = AE' = AE$, we find that $[AE'F] = \frac 12 (5\sqrt{6} + 5\sqrt{2})^2 = 100 + 50\sqrt{3}$.

Now, we need to find $[AED] = [AE'D]$. The acute angles of the triangles are $\frac{15}{2}$ and $90 - \frac{15}{2}$. By repeated application of the half-angle formula, we can find that $\tan \frac{15}{2} = \sqrt{2} - \sqrt{3} + \sqrt{6} - 2$.

The area of $[AED] = \frac 12 \left(20 \cos 15\right)^2 \left(\tan \frac{15}{2}\right)$. Thus, $[AED] + [AE'D] = 2\left(\frac 12((5\sqrt{6} + 5\sqrt{2})^2 \cdot (\sqrt{2} - \sqrt{3} + \sqrt{6} - 2))\right)$, which eventually simplifies to $500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600$.

Adding them together, we find that the solution is $[AEDF] = [AE'F] + [AED] + [AE'D]$ $= 100 + 50\sqrt{3} + 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600=$ $= 500\sqrt{2} - 350\sqrt{3} + 300\sqrt{6} - 600$, and the answer is $\boxed{875}$.

Solution 4

From the given information, calculate the coordinates of all the points (by finding the equations of the lines, equating). Then, use the shoelace method to calculate the area of the intersection.

  • $\overline{AC}$: $y = (\tan 75) x = (2 + \sqrt{3})x$
  • $\overline{AB'}$: $y = (\tan 15) x = (2 - \sqrt{3})x$
  • $\overline{BC}$: It passes thru $(20,0)$, and has a slope of $-\tan75 = -(2 + \sqrt{3})$. The equation of its line is $y = (2+\sqrt{3})(20 - x)$.
  • $\overline{B'C'}$: $AC' = AC = \frac{10}{\cos 75} = 10\sqrt{6} + 10\sqrt{2}$, so it passes thru point $(0, 10\sqrt{6} + 10\sqrt{2})$. It has a slope of $-\tan 60 = -\sqrt{3}$. So the equation of its line is $y = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})$.

Now, we can equate the equations to find the intersections of all the points.

  • $A (0,0)$
  • $D$ is the intersection of $\overline{BC},\ \overline{B'C'}$. $(2+\sqrt{3})(20-x) = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})$. Therefore, $x = 5(4 + 2\sqrt{3}-\sqrt{6}-\sqrt{2})$, $y = 5(3\sqrt{6} + 5\sqrt{2}-4\sqrt{3}-6)$.
  • $E$ is the intersection of $\overline{AB'},\ \overline{BC}$. $(2 - \sqrt{3})x =(2+\sqrt{3})(20-x)$. Therefore, $x = 5(2+\sqrt{3})$, $y = 5$.
  • $F$ is the intersection of $\overline{AC},\ \overline{B'C'}$. $(2+\sqrt{3})x = -\sqrt{3}x + 10(\sqrt{6} + \sqrt{2})$. Therefore, $x = 5\sqrt{2}$, $y = 10\sqrt{2}+ 5\sqrt{6}$.

We take these points and tie them together by shoelace, and the answer should come out to be $\boxed{875}$.

See also

2007 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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