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Difference between revisions of "2007 AIME II Problems/Problem 3"

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(→‎Solution)
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== Solution ==
 
== Solution ==
  
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=== Solution 1 ===
+
===Solution 1===
 +
 
 +
Let <math>\angle FCD = \alpha</math>, so that <math>FB = \sqrt{12^2 + 13^2 + 2\cdot12\cdot13\sin(\alpha)} = \sqrt{433}</math>. By the diagonal, <math>DB = 13\sqrt{2}, DB^2 = 338</math>.
 +
 
 +
<b>The sum of the squares of the sides of a parallelogram is the sum of the squares of the diagonals.</b>
 +
<cmath>EF^2 = 2\cdot(5^2 + 433) - 338 = 578.</cmath>
 +
 
 +
=== Solution 2 ===
  
 
Extend <math>\overline{AE}, \overline{DF}</math> and <math>\overline{BE}, \overline{CF}</math> to their points of intersection. Since <math>\triangle ABE \cong \triangle CDF</math> and are both <math>5-12-13</math> [[right triangle]]s, we can come to the conclusion that the two new triangles are also congruent to these two (use [[ASA]], as we know all the sides are <math>13</math> and the angles are mostly complementary). Thus, we create a [[square]] with sides <math>5 + 12 = 17</math>.
 
Extend <math>\overline{AE}, \overline{DF}</math> and <math>\overline{BE}, \overline{CF}</math> to their points of intersection. Since <math>\triangle ABE \cong \triangle CDF</math> and are both <math>5-12-13</math> [[right triangle]]s, we can come to the conclusion that the two new triangles are also congruent to these two (use [[ASA]], as we know all the sides are <math>13</math> and the angles are mostly complementary). Thus, we create a [[square]] with sides <math>5 + 12 = 17</math>.
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<math>\overline{EF}</math> is the diagonal of the square, with length <math>17\sqrt{2}</math>; the answer is <math>EF^2 = (17\sqrt{2})^2 = 578</math>.
 
<math>\overline{EF}</math> is the diagonal of the square, with length <math>17\sqrt{2}</math>; the answer is <math>EF^2 = (17\sqrt{2})^2 = 578</math>.
  
−
=== Solution 2 ===
+
=== Solution 3 ===
 
A slightly more analytic/brute-force approach:
 
A slightly more analytic/brute-force approach:
  
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Now, we see that <math>FK=DC-(DJ+IB)=DC-2DJ=13-\frac{50}{13}=\frac{119}{13}</math>. Also, <math>EK=BC+(JF+IE)=BC+2JF=13+\frac{120}{13}=\frac{289}{13}</math>. By the Pythagorean Theorem, we have <math>EF=\sqrt{\left(\frac{289}{13}\right)^2+\left(\frac{119}{13} \right)^2}=\frac{\sqrt{(17^2)(17^2+7^2)}}{13}</math><math>=\frac{17\sqrt{338}}{13}=\frac{17(13\sqrt{2})}{13}=17\sqrt{2}</math>. Therefore, <math>EF^2=(17\sqrt{2})^2=578</math>.
 
Now, we see that <math>FK=DC-(DJ+IB)=DC-2DJ=13-\frac{50}{13}=\frac{119}{13}</math>. Also, <math>EK=BC+(JF+IE)=BC+2JF=13+\frac{120}{13}=\frac{289}{13}</math>. By the Pythagorean Theorem, we have <math>EF=\sqrt{\left(\frac{289}{13}\right)^2+\left(\frac{119}{13} \right)^2}=\frac{\sqrt{(17^2)(17^2+7^2)}}{13}</math><math>=\frac{17\sqrt{338}}{13}=\frac{17(13\sqrt{2})}{13}=17\sqrt{2}</math>. Therefore, <math>EF^2=(17\sqrt{2})^2=578</math>.
  
−
===Solution 3===
+
===Solution 4===
 
Based on the symmetry, we know that <math>F</math> is a reflection of <math>E</math> across the center of the square, which we will denote as <math>O</math>. Since <math>\angle BEA</math> and <math>\angle AOB</math> are right, we can conclude that figure <math>AOBE</math> is a cyclic quadrilateral. Pythagorean Theorem yields that <math>BO=AO=\frac{13\sqrt{2}}{2}</math>. Now, using Ptolemy's Theorem, we get that  
 
Based on the symmetry, we know that <math>F</math> is a reflection of <math>E</math> across the center of the square, which we will denote as <math>O</math>. Since <math>\angle BEA</math> and <math>\angle AOB</math> are right, we can conclude that figure <math>AOBE</math> is a cyclic quadrilateral. Pythagorean Theorem yields that <math>BO=AO=\frac{13\sqrt{2}}{2}</math>. Now, using Ptolemy's Theorem, we get that  
 
<cmath>AO\cdot BE + BO\cdot AE = AB\cdot AO</cmath>
 
<cmath>AO\cdot BE + BO\cdot AE = AB\cdot AO</cmath>
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<cmath>OE=\frac{17\sqrt{2}}{2}</cmath>
 
<cmath>OE=\frac{17\sqrt{2}}{2}</cmath>
 
Now, since we stated in the first step that <math>F</math> is a reflection of <math>E</math> across <math>O</math>, we can say that <math>EF=2EO=17\sqrt{2}</math>. This gives that <cmath>EF^2=(17\sqrt{2})^2=578</cmath> AWD with this bash solution
 
Now, since we stated in the first step that <math>F</math> is a reflection of <math>E</math> across <math>O</math>, we can say that <math>EF=2EO=17\sqrt{2}</math>. This gives that <cmath>EF^2=(17\sqrt{2})^2=578</cmath> AWD with this bash solution
−
 
−
===Solution 3===
 
−
Let <math>\angle FCD = \alpha</math>, so that <math>FB = \sqrt{12^2 + 13^2 + 2\cdot12\cdot13\sin(\alpha)} = \sqrt{433}</math>. By the diagonal, <math>DB = 13\sqrt{2}, DB^2 = 338</math>.
 
−
 
−
<b>The sum of the squares of the sides of a parallelogram is the sum of the squares of the diagonals.</b>
 
−
<cmath>EF^2 = 2\cdot(5^2 + 433) - 338 = 578.</cmath>
 
  
 
== See also ==
 
== See also ==

Revision as of 19:12, 9 July 2017

Problem

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5*12/13); draw(A--B--C--D--cycle); draw(A--E--B); draw(C--F--D); dot("$A$", A, W); dot("$B$", B, dir(0)); dot("$C$", C, dir(0)); dot("$D$", D, W); dot("$E$", E, N); dot("$F$", F, S);[/asy]

Solution

Solution 1

Let $\angle FCD = \alpha$, so that $FB = \sqrt{12^2 + 13^2 + 2\cdot12\cdot13\sin(\alpha)} = \sqrt{433}$. By the diagonal, $DB = 13\sqrt{2}, DB^2 = 338$.

The sum of the squares of the sides of a parallelogram is the sum of the squares of the diagonals. \[EF^2 = 2\cdot(5^2 + 433) - 338 = 578.\]

Solution 2

Extend $\overline{AE}, \overline{DF}$ and $\overline{BE}, \overline{CF}$ to their points of intersection. Since $\triangle ABE \cong \triangle CDF$ and are both $5-12-13$ right triangles, we can come to the conclusion that the two new triangles are also congruent to these two (use ASA, as we know all the sides are $13$ and the angles are mostly complementary). Thus, we create a square with sides $5 + 12 = 17$.

[asy]unitsize(0.25 cm); pair A, B, C, D, E, F, G, H; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5*12/13); G = rotate(90,(A + C)/2)*(E); H = rotate(90,(A + C)/2)*(F); draw(A--B--C--D--cycle); draw(E--G--F--H--cycle); dot("$A$", A, N); dot("$B$", B, dir(0)); dot("$C$", C, S); dot("$D$", D, W); dot("$E$", E, N); dot("$F$", F, S); dot("$G$", G, W); dot("$H$", H, dir(0));[/asy]

$\overline{EF}$ is the diagonal of the square, with length $17\sqrt{2}$; the answer is $EF^2 = (17\sqrt{2})^2 = 578$.

Solution 3

A slightly more analytic/brute-force approach:

AIME II prob10 bruteforce.PNG

Drop perpendiculars from $E$ and $F$ to $I$ and $J$, respectively; construct right triangle $EKF$ with right angle at K and $EK || BC$. Since $2[CDF]=DF*CF=CD*JF$, we have $JF=5\times12/13 = \frac{60}{13}$. Similarly, $EI=\frac{60}{13}$. Since $\triangle DJF \sim \triangle DFC$, we have $DJ=\frac{5JF}{12}=\frac{25}{13}$.

Now, we see that $FK=DC-(DJ+IB)=DC-2DJ=13-\frac{50}{13}=\frac{119}{13}$. Also, $EK=BC+(JF+IE)=BC+2JF=13+\frac{120}{13}=\frac{289}{13}$. By the Pythagorean Theorem, we have $EF=\sqrt{\left(\frac{289}{13}\right)^2+\left(\frac{119}{13} \right)^2}=\frac{\sqrt{(17^2)(17^2+7^2)}}{13}$$=\frac{17\sqrt{338}}{13}=\frac{17(13\sqrt{2})}{13}=17\sqrt{2}$. Therefore, $EF^2=(17\sqrt{2})^2=578$.

Solution 4

Based on the symmetry, we know that $F$ is a reflection of $E$ across the center of the square, which we will denote as $O$. Since $\angle BEA$ and $\angle AOB$ are right, we can conclude that figure $AOBE$ is a cyclic quadrilateral. Pythagorean Theorem yields that $BO=AO=\frac{13\sqrt{2}}{2}$. Now, using Ptolemy's Theorem, we get that \[AO\cdot BE + BO\cdot AE = AB\cdot AO\] \[\frac{13\sqrt{2}}{2}\cdot 5+\frac{13\sqrt{2}}{2}\cdot 12 = 13\cdot OE\] \[OE=\frac{17\sqrt{2}}{2}\] Now, since we stated in the first step that $F$ is a reflection of $E$ across $O$, we can say that $EF=2EO=17\sqrt{2}$. This gives that \[EF^2=(17\sqrt{2})^2=578\] AWD with this bash solution

See also

2007 AIME II (Problems • Answer Key • Resources)
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