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

Revision as of 15:06, 22 September 2021

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 $m+n$ ?

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

Solution 1.

Let $BE=a$ , $AD=b$ , and $AC=CE=BD=c$ . $[asy] size(200); defaultpen(linewidth(0.4)+fontsize(10)); pen s = linewidth(0.8)+fontsize(8); pair O,A,B,C,D,E0,F; O=origin; A=(-1,0); path c = CR(O,1); real r = 0.13535; B = IP(c, CR(A,3*r)); C = IP(c, CR(B,10*r)); D = IP(c, CR(C,3*r)); E0 = OP(c, CR(D,10*r)); F = foot(C,A,E0); dot("$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]$ We use cosine formula in $\triangle ABC$ to get $60\cos\theta = 109-c^2$ . Let $F$ be on $AE$ such that $CF \perp AE$ . In $\triangle CFE$ we have $\cos\theta = -\cos(\pi-\theta)=-7/c$ . 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 quadrilateral $ABCE$ to get $a=44/3$ . For $AD=b$ , we use Ptolemy's theorem on quadrilateral $ACDE$ to get $b=27/2$ .

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

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

Revision as of 15:06, 22 September 2021

Contents

Problem

Solution 1.

Solution 2.

See also

@@ Line 8: / Line 8: @@
 \textbf{(E) }421\qquad</math>
-==Solution==
+==Solution 1.==
+Let <math>BE=a</math>, <math>AD=b</math>, and <math>AC=CE=BD=c</math>.
+<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=(-1,0);
+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>
+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
+<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>.
+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.==
 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: