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1979 AHSME Problems/Problem 23

Revision as of 21:35, 21 June 2018 by Wannabecharmander (talk | contribs) (→‎Solution)
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Problem 23

The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.


[asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern("hatch")); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); //Credit to TheMaskedMagician for the diagram[/asy]


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

Solution 1

Note that the distance $PQ$ will be minimized when $P$ is the midpoint of $AB$ and $Q$ is the midpoint of $CD$.

To find this distance, consider triangle $\triangle PCQ$. $Q$ is the midpoint of $CD$, so $CQ=\frac{1}{2}$. Additionally, since $CP$ is the altitude of equilateral $\triangle ABC$, $CP=\frac{\sqrt{3}}{2}$.

[asy] size(150); import patterns; import olympiad; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; draw(BB--DD); P=midpoint(AA--BB); Q=midpoint(CC--DD); draw(P--Q,dashed); draw(P--CC,dashed); draw(AA--CC,dashed); dot(P); dot(Q); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); //Credit to TheMaskedMagician for the diagram //Changes made by Treetor10145[/asy]

Next, we need to find $\cos(\angle PCQ)$ in order to find $PQ$ by the Law of Cosines. To do so, drop down $D$ onto $\triangle ABC$ to get the point $D^\prime$.

$\angle PCD$ is congruent to $\angle D^\prime CD$, since $P$, $D^\prime$, and $C$ are collinear. Therefore, we can just find $\cos(\angle D^\prime CD)$.

Note that $\triangle CD^\prime D$ is a right triangle with $\angle CD^\prime D$ as a right angle.


[asy] size(150); import patterns; import olympiad; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux,R; add("hatch",hatch()); //AA=new A and etc. draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; draw(BB--DD); P=midpoint(AA--BB); Q=midpoint(CC--DD); R=midpoint(AA--CC); pair X=intersectionpoints(P--CC,BB--R)[0]; draw(AA--CC,dashed); draw(DD--X,dashed); draw(X--CC,dashed); draw(rightanglemark(CC,X,DD)); dot(P); dot(Q); dot(X); label("$A$",AA,W); label("$B$",BB,S); label("$C$",CC,E); label("$D$",DD,N); label("$P$",P,S); label("$Q$",Q,E); label("$D^\prime$",X,W); //Credit to TheMaskedMagician for the diagram //Changes made by Treetor10145[/asy]


As given by the problem, $CD=1$.

Note that $D^\prime$ is the centroid of equilateral $\triangle ABC$. Additionally, since $\triangle ABC$ is equilateral, $D^\prime$ is also the orthocenter. Due to this, the distance from $C$ to $D^\prime$ is $\frac{2}{3}$ of the altitude of $\triangle ABC$. Therefore, $CD^\prime=\frac{\sqrt{3}}{3}$.

Since $\cos(\angle D^\prime CD)=\cos(\angle PCQ)=\frac{CD^\prime}{CD}$, $\cos(\angle PCQ)=\frac{\frac{\sqrt{3}}{3}}{1}=\frac{\sqrt{3}}{3}$

\[PQ^2=CP^2+CQ^2-2(CP)(CQ)\cos(\angle PCQ)\] \[PQ^2=\frac{3}{4}+\frac{1}{4}-2\left(\frac{\sqrt{3}}{4}\right)\left(\frac{1}{2}\right)\left(\frac{\sqrt{3}}{3}\right)\] Simplifying, $PQ^2=\frac{1}{2}$. Therefore, $PQ=\frac{\sqrt{2}}{2}\Rightarrow$ $\boxed{\textbf{C}}$

Solution by treetor10145

Solution 2 (less overkill)

Notice, like above said, that $P$ is the midpoint of $AB$ and $Q$ is the midpoint of $CD$.


To find the length of $PQ$, first draw in lines $CP$ and $DP$. Notice that $DP$ is an altitude of $\triangle ADP$. We find that $\angle{DAP} = 60 ^{\circ}$ (since $\triangle ABD$ is equilateral), and $AD=\frac{1}{2}$. Use the properties of 30-60-90 triangles to get $DP=\frac{\sqrt{3}}{2}$. Since $CP$ is an altitude of a congruent equilateral triangle, $CP=DP=\frac{\sqrt{3}}{2}$.


Notice that $\triangle CDP$ is isosceles with $CP=DP$. Also, since $Q$ is the midpoint of base $CD$, we can conclude that $PQ$ is an altitude. We can use Pythagorean theorem to get the following (taking into consideration $DQ=\frac{1}{2}$):


\[DQ^2+PQ^2=PD^2\]


\[\left(\frac{1}{2}\right)^2+PQ^2 = \left(\frac{\sqrt{3}}{2}\right)^2\]


\[PQ^2=\frac{3}{4}-\frac{1}{4}=\frac{1}{2}\]


\[PQ=\frac{\sqrt{2}}{2}\Rightarrow \boxed{\textbf{C}}\]


-WannabeCharmander

See also

1979 AHSME (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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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