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

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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

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 $\overline{AD} = x$ and $\overline{BD} = y$ . 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 equation 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 $2a = \sqrt{2}$

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 33: / Line 33: @@
 === Solution 3 ===
-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>.
+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} = x</math> and <math>\overline{BD} = y</math>.
 Using the Pythagorean Theorem, we obtain the equations

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