Difference between revisions of "2012 AMC 10A Problems/Problem 21"

Latest revision as of 13:09, 26 November 2024

Problem

Let points $A$ = $(0 ,0 ,0)$ , $B$ = $(1, 0, 0)$ , $C$ = $(0, 2, 0)$ , and $D$ = $(0, 0, 3)$ . Points $E$ , $F$ , $G$ , and $H$ are midpoints of line segments $\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. What is the area of $EFGH$ ?

$\textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{3}\qquad\textbf{(C)}\ \frac{3\sqrt{5}}{4}\qquad\textbf{(D)}\ \sqrt{3}\qquad\textbf{(E)}\ \frac{2\sqrt{7}}{3}$

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Solution 1

Consider a tetrahedron with vertices at $A,B,C,D$ in the $xyz$ -space. The length of $EF$ is just one-half of $AD$ because it is the midsegment of $\triangle ABD.$ The same concept applies to the other side lengths. $AD=3$ and $BC=\sqrt{1^2+2^2}=\sqrt{5}$ . Then $EF=HG=\frac32$ and $EH=FG=\frac{\sqrt{5}}{2}$ . The line segments lie on perpendicular planes so quadrilateral $EFGH$ is a rectangle. The area is

$EF \cdot FG = \frac 32 \cdot \frac{\sqrt{5}}{2} = \frac{3\sqrt 5} 4\implies \boxed{\textbf C}.$

$[asy] import three; draw((0,0,0)--(1,0,0)--(0,0,3)--cycle); draw((0,0,0)--(0,2,0)); draw((0,2,0)--(0,0,3)); //EFGH draw((0.5,0,1.5)--(0.5,0,0)--(0,1,0)--(0,1,1.5)--(0.5,0,1.5),red); //Points label("$E$",(0.5,0,1.5),NW); label("$F$",(0.5,0,0),S); label("$G$",(0,1,0),S); label("$H$",(0,1,1.5),NE); label("$A$",(0,0,0),NE); label("$B$",(1,0,0),S); label("$C$",(0,2,0),S); label("$D$",(0,0,3),N); [/asy]$

Solution 2

Computing the points of $EFGH$ gives $E(0.5, 0, 1.5), F(0.5, 0, 0), G(0,1,0), H(0,1,1.5)$ . The vector $EF$ is $(0,0,-1.5)$ , while the vector $HG$ is also $(0,0,-1.5)$ , meaning the two sides $EF$ and $GH$ are parallel. Similarly, the vector $FG$ is $(-0.5, 1, 0)$ , while the vector $EH$ is also $(-0.5, 1, 0)$ . Again, these are equal in both magnitude and direction, so $FG$ and $EH$ are parallel. Thus, figure $EFGH$ is a parallelogram.

Computation of vectors $EF$ and $HG$ is sufficient evidence that the figure is a parallelogram, since the vectors are not only point in the same direction, but are of the same magnitude, but the other vector $FG$ is needed to find the angle between the sides.

Taking the dot product of vector $EF$ and vector $FG$ gives $0 \cdot -0.5 + 0 \cdot 1 + -1.5 \cdot 0 = 0$ , which means the two vectors are perpendicular. (Alternately, as above, note that vector $EF$ goes directly down on the z-axis, while vector $FG$ has no z-component and lie completely in the xy plane.) Thus, the figure is a parallelogram with a right angle, which makes it a rectangle. With the distance formula in three dimensions, we find that $EF = \frac{3}{2}$ and $FG = \frac{\sqrt{5}}{2}$ , giving an area of $\frac32 \cdot \frac{\sqrt{5}}{2} = \boxed{\textbf{(C)}\ \frac{3\sqrt{5}}{4}}$

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012amc10a/249

~dolphin7

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 [[Category: Introductory Geometry Problems]]
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 ==Solution 1==

2012 AMC 10A (Problems • Answer Key • Resources)
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