Difference between revisions of "2014 AMC 12B Problems/Problem 24"

Latest revision as of 07:09, 1 January 2023

Problem

Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$ , $BC = DE = 10$ , and $AE= 14$ . The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

$\textbf{(A) }129\qquad \textbf{(B) }247\qquad \textbf{(C) }353\qquad \textbf{(D) }391\qquad \textbf{(E) }421\qquad$

Video Solution by Punxsutawney Phil

https://www.youtube.com/watch?v=1-2vT_GIceA

Solution 1

Let $BE=a$ , $AD=b$ , and $AC=CE=BD=c$ . Let $F$ be on $AE$ such that $CF \perp AE$ . $[asy] size(200); defaultpen(linewidth(0.4)+fontsize(10)); pen s = linewidth(0.8)+fontsize(8); pair O,A,B,C,D,E0,F; O=origin; A= dir(198); path c = CR(O,1); real r = 0.13535; B = IP(c, CR(A,3*r)); C = IP(c, CR(B,10*r)); D = IP(c, CR(C,3*r)); E0 = OP(c, CR(D,10*r)); F = foot(C,A,E0); dot("$A$", A, A-O); dot("$B$", B, B-O); dot("$C$", C, C-O); dot("$D$", D, D-O); dot("$E$", E0, E0-O); dot("$F$", F, F-C); label("$c$",A--C,S); label("$c$",E0--C,W); label("$7$",F--E0,S); label("$7$",F--A,S); label("$3$",A--B,2*W); label("$10$",B--C,2*N); label("$3$",C--D,2*NE); label("$10$",D--E0,E); draw(A--B--C--D--E0--A, black+0.8); draw(CR(O,1), s); draw(A--C--E0, royalblue); draw(C--F, royalblue+dashed); draw(rightanglemark(E0,F,C,2)); MA("\theta",A,B,C,0.075); MA("\pi-\theta",C,E0,A,0.1); [/asy]$ In $\triangle CFE$ we have $\cos\theta = -\cos(\pi-\theta)=-7/c$ . We use the Law of Cosines on $\triangle ABC$ to get $60\cos\theta = 109-c^2$ . Eliminating $\cos\theta$ we get $c^3-109c-420=0$ which factorizes as $(c+7)(c+5)(c-12)=0.$ Discarding the negative roots we have $c=12$ . Thus $BD=AC=CE=12$ . For $BE=a$ , we use Ptolemy's theorem on cyclic quadrilateral $ABCE$ to get $a=44/3$ . For $AD=b$ , we use Ptolemy's theorem on cyclic quadrilateral $ACDE$ to get $b=27/2$ .

The sum of the lengths of the diagonals is $12+12+12+\tfrac{44}{3}+\tfrac{27}{2} = \tfrac{385}{6}$ so the answer is $385 + 6 = \fbox{\textbf{(D) }391}$

Solution 2

Let $a$ denote the length of a diagonal opposite adjacent sides of length $14$ and $3$ , $b$ for sides $14$ and $10$ , and $c$ for sides $3$ and $10$ . Using Ptolemy's Theorem on the five possible quadrilaterals in the configuration, we obtain:

$\begin{align} c^2 &= 3a+100 \\ c^2 &= 10b+9 \\ ab &= 30+14c \\ ac &= 3c+140\\ bc &= 10c+42 \end{align}$

Using equations $(1)$ and $(2)$ , we obtain:

$a = \frac{c^2-100}{3}$

and

$b = \frac{c^2-9}{10}$

Plugging into equation $(4)$ , we find that:

$\begin{align*} \frac{c^2-100}{3}c &= 3c + 140\\ \frac{c^3-100c}{3} &= 3c + 140\\ c^3-100c &= 9c + 420\\ c^3-109c-420 &=0\\ (c-12)(c+7)(c+5)&=0 \end{align*}$

Or similarly into equation $(5)$ to check:

$\begin{align*} \frac{c^2-9}{10}c &= 10c+42\\ \frac{c^3-9c}{10} &= 10c + 42\\ c^3-9c &= 100c + 420\\ c^3-109c-420 &=0\\ (c-12)(c+7)(c+5)&=0 \end{align*}$

$c$ , being a length, must be positive, implying that $c=12$ . In fact, this is reasonable, since $10+3\approx 12$ in the pentagon with apparently obtuse angles. Plugging this back into equations $(1)$ and $(2)$ we find that $a = \frac{44}{3}$ and $b= \frac{135}{10}=\frac{27}{2}$ .

We desire $3c+a+b = 3\cdot 12 + \frac{44}{3} + \frac{27}{2} = \frac{216+88+81}{6}=\frac{385}{6}$ , so it follows that the answer is $385 + 6 = \fbox{\textbf{(D) }391}$

Solution 3 (Ptolemy's but Quicker)

Let us set $x$ to be $AC=BD=CE$ and $y$ to be $BE$ and $z$ to be $AD$ . It follow from applying Ptolemy's Theorem on $ABCD$ to get $x^2=9+10z$ . Applying Ptolemy's on $ACDE$ gives $xz=42+10x$ ; and applying Ptolemy's on $BCDE$ gives $x^2=100+3y$ . So, we have the have the following system of equations:

$\begin{align} x^2 &= 9+10z \\ x^2 &= 100+3y \\ xz &= 42+10x \end{align}$

From $(3)$ , we have $42=(z-10)x$ . Isolating the x gives $x=\dfrac{42}{z-10}$ . By setting $(1)$ and $(2)$ equal, we have $x^2=9+10z=100+3y$ . Manipulating it gives $3y=10z-91$ . Finally, plugging back into $(2)$ gives $x^2=100+10z-91=10z+9$ . Plugging in the $x=\dfrac{42}{z-10}$ as well gives

$\begin{align*} \left(\frac{42}{z-10}\right)^2 &= 10z+9\\ 10z^3 - 191z^2 + 820z + 900 &= 1764\\ 10z^3 - 191z^2 + 820z - 864 &= 0\\ (5z-8)(2z-27)(z-4) &=0 \end{align*}$

It is impossible for $z<10$ for $x<0$ ; that means $z=\frac{27}{2}$ . That means $x = 12$ and $y = \frac{44}{3}$ .

Thus, the sum of all diagonals is $3x+y+z = 3\cdot 12 + \frac{44}{3} + \frac{27}{2} = 385/6$ , which implies our answer is $m+n = 385+6 = \fbox{391 \textbf{(D)}}$ .

~sml1809

Solution 4

Let $BE = a$ , $AC = CE = BD = b$

By Ptolemy's theorem for quadrilateral $ABCE$ , $AB \cdot CE + BC \cdot AE = BE \cdot AC$ , $3b + 140 = ab$ , $a = 3 + \frac{140}{b}$

By Ptolemy's theorem for quadrilateral $BCDE$ , $CD \cdot BE + BC \cdot DE = BD \cdot CE$ , $3a + 100 = b^2$

$3(3 + \frac{140}{b}) + 100 = b^2$ , $b^3 - 109 b -420 = 0$ , $(b-12)(b+7)(b+5) = 0$ , $b = 12$

$a = 3 + \frac{140}{12} = \frac{44}{3}$

By Ptolemy's theorem for quadrilateral $ABDE$ , $AE \cdot BD + AB \cdot DE = AD \cdot BE$ , $AD \cdot a = 14b + 30$ , $AD = \frac{27}{2}$

$\frac{m}{n} = 12 + 12 + 12 + \frac{44}{3} + \frac{27}{2} = \frac{385}{6}$ , $385 + 6 = \boxed{\textbf{(D) }391}$

~isabelchen

@@ Line 8: / Line 8: @@
 \textbf{(E) }421\qquad</math>
-==Solution 1.==
+==Video Solution by Punxsutawney Phil==
-Let <math>BE=a</math>, <math>AD=b</math>, and <math>AC=CE=BD=c</math>.
+https://www.youtube.com/watch?v=1-2vT_GIceA
+== Solution 1 ==
+Let <math>BE=a</math>, <math>AD=b</math>, and <math>AC=CE=BD=c</math>. Let <math>F</math> be on <math>AE</math> such that <math>CF \perp AE</math>.
 <asy>
 size(200);
@@ Line 17: / Line 20: @@
 pair O,A,B,C,D,E0,F;
 O=origin;
-A=(-1,0);
+A= dir(198);
 path c = CR(O,1);
 real r = 0.13535;
@@ Line 49: / Line 52: @@
 MA("\pi-\theta",C,E0,A,0.1);
 </asy>
-We use cosine formula in <math>\triangle ABC</math> to get <math>60\cos\theta = 109-c^2</math>. Let <math>F</math> be on <math>AE</math> such that <math>CF \perp AE</math>. In <math>\triangle CFE</math> we have <math>\cos\theta = -\cos(\pi-\theta)=-7/c</math>. Eliminating <math>\cos\theta</math> we get <math>c^3-109c-420=0</math> which factorizes as
+In <math>\triangle CFE</math> we have <math>\cos\theta = -\cos(\pi-\theta)=-7/c</math>. We use the [[Law of Cosines]] on <math>\triangle ABC</math> to get <math>60\cos\theta = 109-c^2</math>.  Eliminating <math>\cos\theta</math> we get <math>c^3-109c-420=0</math> which factorizes as
-<cmath>(c+7)(c+5)(c-12)=0.</cmath>Discarding the negative roots we have <math>c=12</math>. Thus <math>BD=AC=CE=12</math>. For <math>BE=a</math>, we use Ptolemy's theorem on quadrilateral <math>ABCE</math> to get <math>a=44/3</math>. For <math>AD=b</math>, we use Ptolemy's theorem on quadrilateral <math>ACDE</math> to get <math>b=27/2</math>.
+<cmath>(c+7)(c+5)(c-12)=0.</cmath>Discarding the negative roots we have <math>c=12</math>. Thus <math>BD=AC=CE=12</math>. For <math>BE=a</math>, we use Ptolemy's theorem on cyclic quadrilateral <math>ABCE</math> to get <math>a=44/3</math>. For <math>AD=b</math>, we use [[Ptolemy's theorem]] on cyclic quadrilateral <math>ACDE</math> to get <math>b=27/2</math>.
-The sum of the lengths of the diagonals is <math>12+12+12+\frac{44}{3}+\frac{27}{2} = \frac{385}{6}</math> so the answer is <math>385 + 6 = \fbox{\textbf{(D) }391}</math>
-==Solution 2.==
+The sum of the lengths of the diagonals is <math>12+12+12+\tfrac{44}{3}+\tfrac{27}{2} = \tfrac{385}{6}</math> so the answer is <math>385 + 6 = \fbox{\textbf{(D) }391}</math>
+== Solution 2 ==
 Let <math>a</math> denote the length of a diagonal opposite adjacent sides of length <math>14</math> and <math>3</math>, <math>b</math> for sides <math>14</math> and <math>10</math>, and <math>c</math> for sides <math>3</math> and <math>10</math>. Using Ptolemy's Theorem on the five possible quadrilaterals in the configuration, we obtain:
@@ Line 107: / Line 109: @@
 We desire <math>3c+a+b = 3\cdot 12 + \frac{44}{3} + \frac{27}{2} = \frac{216+88+81}{6}=\frac{385}{6}</math>, so it follows that the answer is <math>385 + 6 = \fbox{\textbf{(D) }391}</math>
+==Solution 3 (Ptolemy's but Quicker)==
+Let us set <math>x</math> to be <math>AC=BD=CE</math> and <math>y</math> to be <math>BE</math> and <math>z</math> to be <math>AD</math>. It follow from applying [[Ptolemy's Theorem]] on <math>ABCD</math> to get <math>x^2=9+10z</math>. Applying Ptolemy's on <math>ACDE</math> gives <math>xz=42+10x</math>; and applying Ptolemy's on <math>BCDE</math> gives <math>x^2=100+3y</math>. So, we have the have the following system of equations:
+<cmath>
+\begin{align}
+x^2 &= 9+10z \\
+x^2 &= 100+3y \\
+xz &= 42+10x
+\end{align}
+</cmath>
+From <math>(3)</math>, we have <math>42=(z-10)x</math>. Isolating the x gives <math>x=\dfrac{42}{z-10}</math>. By setting <math>(1)</math> and <math>(2)</math> equal, we have <math>x^2=9+10z=100+3y</math>. Manipulating it gives <math>3y=10z-91</math>. Finally, plugging back into <math>(2)</math> gives <math>x^2=100+10z-91=10z+9</math>. Plugging in the <math>x=\dfrac{42}{z-10}</math> as well gives
+<cmath>
+\begin{align*}
+\left(\frac{42}{z-10}\right)^2 &= 10z+9\\
+z^3 - 191z^2 + 820z + 900 &= 1764\\
+z^3 - 191z^2 + 820z - 864 &= 0\\
+(5z-8)(2z-27)(z-4) &=0
+\end{align*}
+</cmath>
+It is impossible for <math>z<10</math> for <math>x<0</math>; that means <math>z=\frac{27}{2}</math>. That means <math>x = 12</math> and <math>y = \frac{44}{3}</math>.
+Thus, the sum of all diagonals is <math>3x+y+z = 3\cdot 12 + \frac{44}{3} + \frac{27}{2} = 385/6</math>, which implies our answer is <math>m+n = 385+6 = \fbox{391 \textbf{(D)}}</math>.
+~sml1809
+==Solution 4==
+[[File:2014AMC12BProblem24Solution4.png|center|400px]]
+Let <math>BE = a</math>, <math>AC = CE = BD = b</math>
+By [[Ptolemy's theorem]] for quadrilateral <math>ABCE</math>, <math>AB \cdot CE + BC \cdot AE = BE \cdot AC</math>, <math>3b + 140 = ab</math>, <math>a = 3 + \frac{140}{b}</math>
+By [[Ptolemy's theorem]] for quadrilateral <math>BCDE</math>, <math>CD \cdot BE + BC \cdot DE = BD \cdot CE</math>, <math>3a + 100 = b^2</math>
+<math>3(3 + \frac{140}{b}) + 100 = b^2</math>, <math>b^3 - 109 b -420 = 0</math>, <math>(b-12)(b+7)(b+5) = 0</math>, <math>b = 12</math>
+<math>a = 3 + \frac{140}{12} = \frac{44}{3}</math>
+By [[Ptolemy's theorem]] for quadrilateral <math>ABDE</math>, <math>AE \cdot BD + AB \cdot DE = AD \cdot BE</math>, <math>AD \cdot a = 14b + 30</math>, <math>AD = \frac{27}{2}</math>
+<math>\frac{m}{n} = 12 + 12 + 12 + \frac{44}{3} + \frac{27}{2} = \frac{385}{6}</math>, <math>385 + 6 = \boxed{\textbf{(D) }391}</math>
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 == See also ==

2014 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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