Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2005 AMC 10B Problems/Problem 14"

(→‎Solution 4)
m (→‎Solution 5)
 
(17 intermediate revisions by 4 users not shown)
Line 14: Line 14:
 
draw(C--D--M--cycle);</asy>
 
draw(C--D--M--cycle);</asy>
  
−
<math> \textrm{(A)}\ \frac {\sqrt {2}}{2}\qquad \textrm{(B)}\ \frac {3}{4}\qquad \textrm{(C)}\ \frac {\sqrt {3}}{2}\qquad \textrm{(D)}\ 1\qquad \textrm{(E)}\ \sqrt {2}</math>
+
<math> \textbf{(A) }\ \frac {\sqrt {2}}{2}\qquad \textbf{(B) }\ \frac {3}{4}\qquad \textbf{(C) }\ \frac {\sqrt {3}}{2}\qquad \textbf{(D) }\ 1\qquad \textbf{(E) }\ \sqrt {2}</math>
  
 
== Solutions ==
 
== Solutions ==
−
===Solution 1 (simplest) ===
+
===Solution 1 (trig) ===
 
The area of a triangle can be given by <math>\frac12 ab \sin C</math>. <math>MC=1</math> because it is the midpoint of a side, and <math>CD=2</math> because it is the same length as <math>BC</math>. Each angle of an equilateral triangle is <math>60^\circ</math> so <math>\angle MCD = 120^\circ</math>. The area is <math>\frac12 (1)(2) \sin
 
The area of a triangle can be given by <math>\frac12 ab \sin C</math>. <math>MC=1</math> because it is the midpoint of a side, and <math>CD=2</math> because it is the same length as <math>BC</math>. Each angle of an equilateral triangle is <math>60^\circ</math> so <math>\angle MCD = 120^\circ</math>. The area is <math>\frac12 (1)(2) \sin
 
  120^\circ = \boxed{\textbf{(C)}\ \frac{\sqrt{3}}{2}}</math>.
 
  120^\circ = \boxed{\textbf{(C)}\ \frac{\sqrt{3}}{2}}</math>.
−
Note: Even if you don't know the value of <math>\sin 120^\circ</math>, you can use the fact that  <math>\sin x = \sin 180^\circ - x</math>, so <math>\sin 120^\circ = \sin 60^\circ</math>.
+
Note: Even if you don't know the value of <math>\sin 120^\circ</math>, you can use the fact that  <math>\sin x = \sin (180^\circ - x)</math>, so <math>\sin 120^\circ = \sin 60^\circ</math>.
 
You can easily calculate <math>\sin 60^\circ</math> to be <math>\frac{\sqrt3}{2}</math> using equilateral triangles.
 
You can easily calculate <math>\sin 60^\circ</math> to be <math>\frac{\sqrt3}{2}</math> using equilateral triangles.
 +
 +
~Minor Edits by doulai1
 +
 
===Solution 2===
 
===Solution 2===
−
In order to calculate the area of <math>\triangle CDM</math>, we can use the formula <math>A=\dfrac{1}{2}bh</math>, where <math>\overline{CD}</math> is the base. We already know that <math>\overline{CD}=2</math>, so the formula now becomes <math>A=h</math>. We can drop verticals down from <math>A</math> and <math>M</math> to points <math>E</math> and <math>F</math>, respectively. We can see that <math>\triangle AEC \sim \triangle MFC</math>. Now, we establish the relationship that <math>\dfrac{AE}{MF}=\dfrac{AC}{MC}</math>. We are given that <math>\overline{AC}=2</math>, and <math>M</math> is the midpoint of <math>\overline{AC}</math>, so <math>\overline{MC}=1</math>. Because <math>\triangle AEB</math> is a <math>30-60-90</math> triangle and the ratio of the sides opposite the angles are <math>1-\sqrt{3}-2</math> <math>\overline{AE}</math> is <math>\sqrt{3}</math>. Plugging those numbers in, we have <math>\dfrac{\sqrt{3}}{MF}=\dfrac{2}{1}</math>. Cross-multiplying, we see that <math>2\times\overline{MF}=\sqrt{3}\times1\implies \overline{MF}=\dfrac{\sqrt{3}}{2}</math> Since <math>\overline{MF}</math> is the height <math>\triangle CDM</math>, the area is <math>\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}</math>.
+
In order to calculate the area of <math>\triangle CDM</math>, we can use the formula <math>A=\dfrac{1}{2}bh</math>, where <math>\overline{CD}</math> is the base. We already know that <math>\overline{CD}=2</math>, so the formula becomes <math>A=h</math>. We can drop verticals down from <math>A</math> and <math>M</math> to points <math>E</math> and <math>F</math>, respectively. We can see that <math>\triangle AEC \sim \triangle MFC</math>. Now, we establish the relationship that <math>\dfrac{AE}{MF}=\dfrac{AC}{MC}</math>. We are given that <math>\overline{AC}=2</math>, and <math>M</math> is the midpoint of <math>\overline{AC}</math>, so <math>\overline{MC}=1</math>. Because <math>\triangle AEB</math> is a <math>30-60-90</math> triangle and the ratio of the sides opposite the angles are <math>1-\sqrt{3}-2</math> <math>\overline{AE}</math> is <math>\sqrt{3}</math>. Plugging those numbers in, we have <math>\dfrac{\sqrt{3}}{MF}=\dfrac{2}{1}</math>. Cross-multiplying, we see that <math>2\times\overline{MF}=\sqrt{3}\times1\implies \overline{MF}=\dfrac{\sqrt{3}}{2}</math> Since <math>\overline{MF}</math> is the height <math>\triangle CDM</math>, the area is <math>\boxed{\textbf{(C) }\frac{\sqrt{3}}{2}}</math>.
  
 
===Solution 3===
 
===Solution 3===
−
Draw a line from <math>M</math> to the midpoint of <math>\overline{BC}</math>. Call the midpoint of <math>\overline{BC}</math> <math>P</math>. This is an equilateral triangle, since the two segments <math>\overline{PC}</math> and <math>\overline{MC}</math> are identical, and <math>\angle C</math> is 60°. Using the Pythagorean Theorem, point <math>M</math> to <math>\overline{BC}</math> is <math>\dfrac{\sqrt{3}}{2}</math>. Also, the length of <math>\overline{CD}</math> is 2, since <math>C</math> is the midpoint of <math>\overline{BD}</math>. So, our final equation is <math>\dfrac{\sqrt{3}}{2}\times2\over2</math>, which just leaves us with <math>\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}</math>.
+
Draw a line from <math>M</math> to the midpoint of <math>\overline{BC}</math>. Call the midpoint of <math>\overline{BC}</math> <math>P</math>. This is an equilateral triangle, since the two segments <math>\overline{PC}</math> and <math>\overline{MC}</math> are identical, and <math>\angle C</math> is <math>60^{\circ}</math>. Using the [[Pythagorean Theorem]], point <math>M</math> to <math>\overline{BC}</math> is <math>\dfrac{\sqrt{3}}{2}</math>. Also, the length of <math>\overline{CD}</math> is 2, since <math>C</math> is the midpoint of <math>\overline{BD}</math>. So, our final equation is <math>\frac{\sqrt{3}}{2}\times2\over2</math>, which just leaves us with <math>\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}</math>.
  
 
===Solution 4 ===
 
===Solution 4 ===
−
Drop a vertical down from <math>M</math> to <math>BC</math>. Let us call the point of intersection <math>X</math> and the midpoint of <math>BC</math>, <math>Y</math>. We can observe that <math>\triangle AYC</math> and <math>\triangle MXC</math> are similar. By the Pythagorean theorem, <math>AY</math> is <math>\sqrt3</math>. Since <math>AC:MC=2:1,</math> we find <math>MX=\frac{\sqrt3}{2}.</math> Because <math>C</math> is the midpoint of <math>BD,</math> and <math>BC=2,</math> <math>CD=2.</math> Using the area formula, <math>\frac{CD*MX}{2}=\frac{\sqrt3}{2},</math> <math>\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}.</math>
+
Drop a vertical down from <math>M</math> to <math>BC</math>. Let us call the point of intersection <math>X</math> and the midpoint of <math>BC</math>, <math>Y</math>. We can observe that <math>\triangle AYC</math> and <math>\bigtriangleup MXC</math> are similar. By the [[Pythagorean theorem]], <math>AY</math> is <math>\sqrt3</math>.
 +
 
 +
Since <math>AC:MC=2:1,</math> we find <math>MX=\frac{\sqrt3}{2}.</math> Because <math>C</math> is the midpoint of <math>BD,</math> and <math>BC=2,</math> <math>CD=2.</math> Using the area formula, <math>\frac{CD*MX}{2}=\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}.</math>
  
 
~ sdk652
 
~ sdk652
 +
 +
===Solution 5 ===
 +
Think of <math>\triangle ABC</math> and <math>\triangle MCD</math> being independent. Now to find area's we just solve for ratios between the triangles that we can plug in the value of <math>2</math> (for a side of <math>\triangle ABC</math>) for. Looking at the information, we see that <math>C</math> is the midpoint of <math>\overline{BD}</math>, and this means that it bisects <math>\overline{BD}</math> which results in <math>BC=CD</math>. Now for the height, we can see that <math>M</math> is the midpoint of <math>\overline{AC}</math> which means that <math>AM=CM</math>, and in turn means that the height of <math>\triangle MCD</math> is half of that of <math>\triangle ABC</math>, and now plugging the ratios of the bases being the same while the height is half of the other triangle, we end up with the area of <math>\triangle MCD</math> being half of that of <math>\triangle ABC</math>. Now all that's left is to find the area of <math>\triangle ABC</math>, and for that, we plug in <math>2</math> which leads us to the answer of <math>3</math>, but since we need to divide by two, our final answer is <math>\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}.</math>
 +
 +
~<math>\LaTeX</math> help by vadava_lx
  
 
== See Also ==
 
== See Also ==

Latest revision as of 14:15, 23 June 2024

Problem

Equilateral $\triangle ABC$ has side length $2$, $M$ is the midpoint of $\overline{AC}$, and $C$ is the midpoint of $\overline{BD}$. What is the area of $\triangle CDM$? [asy]defaultpen(linewidth(.8pt)+fontsize(8pt)); pair B = (0,0); pair A = 2*dir(60); pair C = (2,0); pair D = (4,0); pair M = midpoint(A--C); label("$A$",A,NW);label("$B$",B,SW);label("$C$",C, SE);label("$M$",M,NE);label("$D$",D,SE); draw(A--B--C--cycle); draw(C--D--M--cycle);[/asy]

$\textbf{(A) }\ \frac {\sqrt {2}}{2}\qquad \textbf{(B) }\ \frac {3}{4}\qquad \textbf{(C) }\ \frac {\sqrt {3}}{2}\qquad \textbf{(D) }\ 1\qquad \textbf{(E) }\ \sqrt {2}$

Solutions

Solution 1 (trig)

The area of a triangle can be given by $\frac12 ab \sin C$. $MC=1$ because it is the midpoint of a side, and $CD=2$ because it is the same length as $BC$. Each angle of an equilateral triangle is $60^\circ$ so $\angle MCD = 120^\circ$. The area is $\frac12 (1)(2) \sin 120^\circ = \boxed{\textbf{(C)}\ \frac{\sqrt{3}}{2}}$. Note: Even if you don't know the value of $\sin 120^\circ$, you can use the fact that $\sin x = \sin (180^\circ - x)$, so $\sin 120^\circ = \sin 60^\circ$. You can easily calculate $\sin 60^\circ$ to be $\frac{\sqrt3}{2}$ using equilateral triangles.

~Minor Edits by doulai1

Solution 2

In order to calculate the area of $\triangle CDM$, we can use the formula $A=\dfrac{1}{2}bh$, where $\overline{CD}$ is the base. We already know that $\overline{CD}=2$, so the formula becomes $A=h$. We can drop verticals down from $A$ and $M$ to points $E$ and $F$, respectively. We can see that $\triangle AEC \sim \triangle MFC$. Now, we establish the relationship that $\dfrac{AE}{MF}=\dfrac{AC}{MC}$. We are given that $\overline{AC}=2$, and $M$ is the midpoint of $\overline{AC}$, so $\overline{MC}=1$. Because $\triangle AEB$ is a $30-60-90$ triangle and the ratio of the sides opposite the angles are $1-\sqrt{3}-2$ $\overline{AE}$ is $\sqrt{3}$. Plugging those numbers in, we have $\dfrac{\sqrt{3}}{MF}=\dfrac{2}{1}$. Cross-multiplying, we see that $2\times\overline{MF}=\sqrt{3}\times1\implies \overline{MF}=\dfrac{\sqrt{3}}{2}$ Since $\overline{MF}$ is the height $\triangle CDM$, the area is $\boxed{\textbf{(C) }\frac{\sqrt{3}}{2}}$.

Solution 3

Draw a line from $M$ to the midpoint of $\overline{BC}$. Call the midpoint of $\overline{BC}$ $P$. This is an equilateral triangle, since the two segments $\overline{PC}$ and $\overline{MC}$ are identical, and $\angle C$ is $60^{\circ}$. Using the Pythagorean Theorem, point $M$ to $\overline{BC}$ is $\dfrac{\sqrt{3}}{2}$. Also, the length of $\overline{CD}$ is 2, since $C$ is the midpoint of $\overline{BD}$. So, our final equation is $\frac{\sqrt{3}}{2}\times2\over2$, which just leaves us with $\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}$.

Solution 4

Drop a vertical down from $M$ to $BC$. Let us call the point of intersection $X$ and the midpoint of $BC$, $Y$. We can observe that $\triangle AYC$ and $\bigtriangleup MXC$ are similar. By the Pythagorean theorem, $AY$ is $\sqrt3$.

Since $AC:MC=2:1,$ we find $MX=\frac{\sqrt3}{2}.$ Because $C$ is the midpoint of $BD,$ and $BC=2,$ $CD=2.$ Using the area formula, $\frac{CD*MX}{2}=\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}.$

~ sdk652

Solution 5

Think of $\triangle ABC$ and $\triangle MCD$ being independent. Now to find area's we just solve for ratios between the triangles that we can plug in the value of $2$ (for a side of $\triangle ABC$) for. Looking at the information, we see that $C$ is the midpoint of $\overline{BD}$, and this means that it bisects $\overline{BD}$ which results in $BC=CD$. Now for the height, we can see that $M$ is the midpoint of $\overline{AC}$ which means that $AM=CM$, and in turn means that the height of $\triangle MCD$ is half of that of $\triangle ABC$, and now plugging the ratios of the bases being the same while the height is half of the other triangle, we end up with the area of $\triangle MCD$ being half of that of $\triangle ABC$. Now all that's left is to find the area of $\triangle ABC$, and for that, we plug in $2$ which leads us to the answer of $3$, but since we need to divide by two, our final answer is $\boxed{\textbf{(C) }\dfrac{\sqrt{3}}{2}}.$

~$\LaTeX$ help by vadava_lx

See Also

2005 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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 10 Problems and Solutions

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10B_Problems/Problem_14&oldid=220975"