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Difference between revisions of "2002 AMC 12B Problems/Problem 23"

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\qquad\mathrm{(D)}\ \frac 32
 
\qquad\mathrm{(D)}\ \frac 32
 
\qquad\mathrm{(E)}\ \sqrt{3}</math>
 
\qquad\mathrm{(E)}\ \sqrt{3}</math>
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== Solution ==
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[[Image:2002_12B_AMC-23.png]]
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== Solution ==
 
== Solution ==
  
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Hence <math>a = \frac{1}{\sqrt{2}}</math> and <math>BC = 2a = \sqrt{2} \Rightarrow \mathrm{(C)}</math>.
 
Hence <math>a = \frac{1}{\sqrt{2}}</math> and <math>BC = 2a = \sqrt{2} \Rightarrow \mathrm{(C)}</math>.
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 +
=== Solution 2 ===
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 +
From [[Stewart's Theorem]], we have <math>(2)(1/2)a(2) + (1)(1/2)a(1) = (a)(a)(a) + (1/2)a(a)(1/2)a.</math> Simplifying, we get <math>(5/4)a^3 = (5/2)a \implies (5/4)a^2 = 5/2 \implies a^2 = 2 \implies a = \boxed{\sqrt{2}}.</math>
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=== Solution 3 ===
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Let <math>D</math> be the foot of the altitude from <math>A</math> to <math>\overline{BC}</math> extended past <math>B</math>. Let <math>\overline{AD}</math> be <math>x</math> and <math>\overline{BD}</math> be <math>y</math>.
 +
Using the Pythagorean Theorem, we obtain the equations
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 +
<cmath>
 +
\begin{align*}
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x^2 + y^2 = 1 \\
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x^2 + y^2 + 2ya + a^2 = 4a^2 \\
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x^2 + y^2 + 4ya + 4a^2 = 4
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\end{align*}
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</cmath>
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 +
Subtracting the first from the second and third equations, we get
 +
 +
<cmath>
 +
\begin{align*}
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2ya + a^2 = 4a^2 - 1 \\
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4ya + 4a^2 = 3
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\end{align*}
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</cmath>
 +
 +
Then subtracting two times the first equation from the second and rearranging, we get <math>2a^2 = 1</math>, so <math>2a = \sqrt{2}</math>
  
 
=== Solution 2 ===
 
=== Solution 2 ===

Revision as of 22:07, 26 November 2017

Problem

In $\triangle ABC$, we have $AB = 1$ and $AC = 2$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$?

$\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 \qquad\mathrm{(C)}\ \sqrt{2} \qquad\mathrm{(D)}\ \frac 32 \qquad\mathrm{(E)}\ \sqrt{3}$

Solution

2002 12B AMC-23.png

Solution

2002 12B AMC-23.png

Solution 1

Let $D$ be the foot of the median from $A$ to $\overline{BC}$, and we let $AD = BC = 2a$. Then by the Law of Cosines on $\triangle ABD, \triangle ACD$, we have \begin{align*} 1^2 &= a^2 + (2a)^2 - 2(a)(2a)\cos ADB \\ 2^2 &= a^2 + (2a)^2 - 2(a)(2a)\cos ADC \end{align*}

Since $\cos ADC = \cos (180 - ADB) = -\cos ADB$, we can add these two equations and get

\[5 = 10a^2\]

Hence $a = \frac{1}{\sqrt{2}}$ and $BC = 2a = \sqrt{2} \Rightarrow \mathrm{(C)}$.

Solution 2

From Stewart's Theorem, we have $(2)(1/2)a(2) + (1)(1/2)a(1) = (a)(a)(a) + (1/2)a(a)(1/2)a.$ Simplifying, we get $(5/4)a^3 = (5/2)a \implies (5/4)a^2 = 5/2 \implies a^2 = 2 \implies a = \boxed{\sqrt{2}}.$

Solution 3

Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$ extended past $B$. Let $\overline{AD}$ be $x$ and $\overline{BD}$ be $y$. Using the Pythagorean Theorem, we obtain the equations

\begin{align*} x^2 + y^2 = 1 \\ x^2 + y^2 + 2ya + a^2 = 4a^2 \\ x^2 + y^2 + 4ya + 4a^2 = 4 \end{align*}

Subtracting the first from the second and third equations, we get

\begin{align*} 2ya + a^2 = 4a^2 - 1 \\ 4ya + 4a^2 = 3 \end{align*}

Then subtracting two times the first equation from the second and rearranging, we get $2a^2 = 1$, so $2a = \sqrt{2}$

Solution 2

From Stewart's Theorem, we have $(2)(1/2)a(2) + (1)(1/2)a(1) = (a)(a)(a) + (1/2)a(a)(1/2)a.$ Simplifying, we get $(5/4)a^3 = (5/2)a \implies (5/4)a^2 = 5/2 \implies a^2 = 2 \implies a = \boxed{\sqrt{2}}.$

See also

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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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