2008 Mock ARML 1 Problems/Problem 3

Problem

In regular hexagon $ABCDEF$ with side length $1$ , $AD$ intersects $BF$ at $G$ , and $BD$ intersects $EC$ at $H$ . Compute the length of $GH$ .

Solution 1

$[asy] pointpen = black; pathpen = black + linewidth(0.62); pair v(int n){ return dir(n * 60); } D(MP("A",v(0))--MP("B",v(1),N)--MP("C",v(2),N)--MP("D",v(3),SW)--MP("E",v(4))--MP("F",v(5))--cycle); D(v(0)--v(3));D(v(1)--v(5));D(v(1)--v(3));D(v(2)--v(4)); D(D(MP("G",IP(v(0)--v(3),v(1)--v(5)),NE))--D(MP("H",IP(v(1)--v(3),v(2)--v(4)),NW)),dashed); D(MP("H'",IP(v(2)--v(4),v(0)--v(3)),SW)); [/asy]$

Let $H'$ be the foot of the perpendicular from $H$ to $\overline{AD}$ . Since $\angle CDA$ is an inscribed angle with measure $\frac{120}{2} = 60^{\circ}$ , it follows that $\triangle CDH'$ is a $30-60-90 \triangle$ , and $DH' = \frac{1}{2}$ and $CH' = BG = \frac{\sqrt{3}}{2}$ . Also, $H'G = CB = 1$ . Note that $\triangle DH'H \sim \triangle DGB$ by ratio $1/3$ . Thus $HH' = BG/3 = \frac{\sqrt{3}}{6}$ .

By the Pythagorean Theorem, $HG^2 = H'H^2 + H'H^2 = \left(\frac{\sqrt{3}}{6}\right)^2 + 1 = \frac{13}{12}$ . Thus $HG = \boxed{\frac{\sqrt{39}}{6}}$ .

Solution 2

We use coordinates. Consider the diagram in Solution 1. Let $D=(0,0)$ . By 30-60-90 triangle ratios, we can get that $C=(\frac12, \frac{\sqrt3}{2})$ , and $H'=(\frac12, 0)$ . In addition, $B=(\frac32, \frac{\sqrt3}{2})$ and $E=(\frac12, -\frac{\sqrt3}{2})$ . The line $\overline{CE}$ is represented by $x=\frac12$ . The line $\overline{BD}$ is represented by $y=\frac{\sqrt3}{3}x$ . The intersection ( $H$ ) is then $(\frac12, \frac{\sqrt3}{6})$ . We additionaly can get that $G=(\frac32,0)$ . Then, by the distance formula, the length of $GH$ is $\sqrt{1^2+\left(\frac{\sqrt3}{6}\right)^2}=\boxed{\frac{\sqrt{39}}{6}}$ .

~yofro

Solution 3 (30 Seconds)

Use the diagram in Solution 1. Notice that $\triangle DHH'\sim \triangle BHC$ . The ratio is $1:2$ , since $DH'=\frac12$ and $BC=1$ . Letting $HH'=x$ , we see that $3x=\frac{\sqrt3}{2}\implies x=\frac{\sqrt3}{6}$ . Then by the Pythagorean Theorem, $GH=\boxed{\frac{\sqrt{39}}{6}}$ .