Difference between revisions of "2002 AMC 12B Problems/Problem 23"
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\qquad\mathrm{(E)}\ \sqrt{3}</math> | \qquad\mathrm{(E)}\ \sqrt{3}</math> | ||
== Solution == | == Solution == | ||
+ | |||
+ | === Solution 1: Pythagorean Theorem === | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | unitsize(4cm); | ||
+ | |||
+ | pair A, B, C, D, M; | ||
+ | |||
+ | A = (1.768,0.935); | ||
+ | B = (1.414,0); | ||
+ | C = (0,0); | ||
+ | D = (1.768,0); | ||
+ | M = (0.707,0); | ||
+ | |||
+ | draw(A--B--C--cycle); | ||
+ | draw(A--D); | ||
+ | draw(D--B); | ||
+ | draw(A--M); | ||
+ | |||
+ | label("$A$",A,N); | ||
+ | label("$B$",B,S); | ||
+ | label("$C$",C,S); | ||
+ | label("$D$",D,S); | ||
+ | label("$M$",M,S); | ||
+ | label("$x$",(A+D)/2,E); | ||
+ | label("$y$",(B+D)/2,S); | ||
+ | label("$a$",(C+M)/2,S); | ||
+ | label("$a$",(M+B)/2,S); | ||
+ | label("$2a$",(A+M)/2,SE); | ||
+ | label("$1$",(A+B)/2,SE); | ||
+ | label("$2$",(A+C)/2,NW); | ||
+ | |||
+ | draw(rightanglemark(B,D,A,3)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | 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>AD = x</math> and <math>BD = y</math>. | ||
+ | Using the Pythagorean Theorem, we obtain the equations | ||
+ | |||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | x^2 + y^2 = 1 \hspace{0.5cm}(1)\\ | ||
+ | x^2 + y^2 + 2ya + a^2 = 4a^2 \hspace{0.5cm}(2)\\ | ||
+ | x^2 + y^2 + 4ya + 4a^2 = 4 \hspace{0.5cm}(3) | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Subtracting <math>(1)</math> equation from <math>(2)</math> and <math>(3)</math>, we get | ||
+ | |||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | 2ya + a^2 = 4a^2 - 1 \hspace{0.5cm}(4)\\ | ||
+ | 4ya + 4a^2 = 3 \hspace{0.5cm}(5) | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Then, subtracting <math>2 \times (4)</math> from <math>(5)</math> and rearranging, we get <math>10a^2 = 5</math>, so <math>BC = 2a = \sqrt{2}\Rightarrow \boxed{\mathrm{(C)}}</math> | ||
+ | |||
+ | ~greenturtle 11/28/2017 | ||
+ | |||
+ | === Solution 2: Law of Cosines === | ||
[[Image:2002_12B_AMC-23.png]] | [[Image:2002_12B_AMC-23.png]] | ||
− | |||
− | |||
Let <math>D</math> be the foot of the median from <math>A</math> to <math>\overline{BC}</math>, and we let <math>AD = BC = 2a</math>. Then by the [[Law of Cosines]] on <math>\triangle ABD, \triangle ACD</math>, we have | Let <math>D</math> be the foot of the median from <math>A</math> to <math>\overline{BC}</math>, and we let <math>AD = BC = 2a</math>. Then by the [[Law of Cosines]] on <math>\triangle ABD, \triangle ACD</math>, we have | ||
Line 27: | Line 87: | ||
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>. | ||
− | === Solution | + | === Solution 3: Stewart's Theorem === |
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> | 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> | ||
+ | - awu2014 | ||
− | === Solution | + | === Solution 4: Pappus's Median Theorem === |
− | + | There is a theorem in geometry known as Pappus's Median Theorem. It states that if you have <math>\triangle{ABC}</math>, and you draw a median from point <math>A</math> to side <math>BC</math> (label this as <math>M</math>), then: <math>(AM)^2 = \dfrac{2(b^2) + 2(c^2) - (a^2)}{4}</math>. Note that <math>b</math> is the length of side <math>\overline{AC}</math>, <math>c</math> is the length of side <math>\overline{AB}</math>, and <math>a</math> is length of side <math>\overline{BC}</math>. Let <math>MB = MC = x</math>. Then <math>AM = 2x</math>. Now, we can plug into the formula given above: <math>AM = 2x</math>, <math>b = 2</math>, <math>c = 1</math>, and <math>a = 2x</math>. After some simple algebra, we find <math>x = \dfrac{\sqrt{2}}{2}</math>. Then, <math>BC = \boxed{\sqrt{2}} \implies \boxed{C}</math>. | |
− | |||
− | + | -Flames | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | Note: Pappus's Median Theorem is just a special case of Stewart's Theorem, with <math>m = n</math>. ~Puck_0 | |
+ | aka Apollonius' Theorem - Orion 2010 | ||
− | + | ===Video Solution by TheBeautyofMath=== | |
− | + | https://youtu.be/jEVMgWKQIW8 | |
− | |||
− | |||
− | |||
− | |||
− | + | ~IceMatrix | |
== See also == | == See also == |
Latest revision as of 12:53, 9 July 2024
Contents
Problem
In , we have and . Side and the median from to have the same length. What is ?
Solution
Solution 1: Pythagorean Theorem
Let be the foot of the altitude from to extended past . Let and . Using the Pythagorean Theorem, we obtain the equations
Subtracting equation from and , we get
Then, subtracting from and rearranging, we get , so
~greenturtle 11/28/2017
Solution 2: Law of Cosines
Let be the foot of the median from to , and we let . Then by the Law of Cosines on , we have
Since , we can add these two equations and get
Hence and .
Solution 3: Stewart's Theorem
From Stewart's Theorem, we have Simplifying, we get - awu2014
Solution 4: Pappus's Median Theorem
There is a theorem in geometry known as Pappus's Median Theorem. It states that if you have , and you draw a median from point to side (label this as ), then: . Note that is the length of side , is the length of side , and is length of side . Let . Then . Now, we can plug into the formula given above: , , , and . After some simple algebra, we find . Then, .
-Flames
Note: Pappus's Median Theorem is just a special case of Stewart's Theorem, with . ~Puck_0 aka Apollonius' Theorem - Orion 2010
Video Solution by TheBeautyofMath
~IceMatrix
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.