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

Revision as of 13:34, 1 January 2020

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$",D,SE); draw(A--B--C--cycle); draw(C--D--M--cycle);[/asy]$ $\textrm{(A)}\ \frac {\sqrt {2}}{2}\qquad \textrm{(B)}\ \frac {3}{4}\qquad \textrm{(C)}\ \frac {\sqrt {3}}{2}\qquad \textrm{(D)}\ 1\qquad \textrm{(E)}\ \sqrt {2}$

Solution

Solution 1

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 $\frac12 (1)(2) \sin 120^\circ = \boxed{\textbf{(C)}\ \frac{\sqrt{3}}{2}}$ .

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 now 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 $\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}$ .

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°. 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 $\dfrac{\sqrt{3}}{2}\times2\over2$ , which just leaves us with $\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}$ .

Solution 4

Drop a vertical down from $M$ to $BC$ . WLOG, let us call the point of intersection $X$ and the midpoint of $BC$ , $Y$ . We can observe that $\triangle AYC$ and $\triangle MXC$ 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, $\frac{CD*MX}{2}=\frac{\sqrt3}{2},$ $\boxed{\mathrm{(C)}\ \dfrac{\sqrt{3}}{2}}.$

sdk652

@@ Line 24: / Line 24: @@
 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>.
-===Solution 4 (Simplest)===
+===Solution 4 ===
 Drop a vertical down from <math>M</math> to <math>BC</math>. WLOG, 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>

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