Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]
size(120);
draw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);
draw((1,0)--(1,3.464));
label("10",(3.5,2.598),NE);
label("$A$",(1,0),SW);
label("$D$",(1,3.464),NW);
[/asy]
Label point $X$ as shown below, and let $Y$ be the foot of the perpendicular from $X$ to $AD$. [asy]

size(120);
pair A,B,C,D,E,F;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300); label("$10$",(A+B)/2,NE);
pair H=(E+C)/2; draw(D--H); draw(E--C); label("$D$",C,NW);label("$X$",D,W);label("$A$",E,SW);label("$Y$",H,E);
draw(A--B--C--D--E--F--A);
[/asy] Since the hexagon is regular, $\angle DXA = 120^\circ$ and $\angle AXY = \angle DXY = 120^\circ / 2 = 60^\circ$.  Thus, $\triangle AXY$ and $\triangle DXY$ are congruent $30^\circ - 60^\circ - 90^\circ$ triangles.  These triangles are each half an equilateral triangle, so their short leg is half as long as their hypotenuse.

Since the side length of the hexagon is 10, we have $AX=XD=10$.  It follows that $XY = AX/2 = 5$ and $AY = DY = \sqrt{10^2-5^2} = \sqrt{75} = 5\sqrt{3}$.  (Notice that this value is $\sqrt{3}$ times the length of $XY$, the short leg.  In general, the ratio of the sides in a $30^\circ - 60^\circ - 90^\circ$ is $1:\sqrt{3}:2$, which can be shown by the Pythagorean Theorem.)  Then, $DA = 2\cdot 5\sqrt{3} = \boxed{10\sqrt{3}}$.
10\sqrt{3}
Level 3
Geometry
Problem: Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]
size(120);
draw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);
draw((1,0)--(1,3.464));
label("10",(3.5,2.598),NE);
label("$A$",(1,0),SW);
label("$D$",(1,3.464),NW);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the length of diagonal $DA$ in the regular hexagon, we first recognize that a regular hexagon can be divided into six equilateral triangles. Each side of the hexagon is equal, and the distance from the center to any vertex is the same as the length of a side.

Given that the side length of the hexagon is 10 units, we can determine the coordinates of points $A$ and $D$. Point $A$ is at the bottom left corner of the hexagon, which can be represented as $(1, 0)$. Point $D$ is directly above $A$, at the top left corner, represented as $(1, 3\sqrt{3})$.

To find the length of diagonal $DA$, we apply the distance formula between points $A(1, 0)$ and $D(1, 3\sqrt{3})$. The distance formula is given by:

Distance = √[(x2 - x1)² + (y2 - y1)²]

Substituting the coordinates of points $A$ and $D$ into the formula:

Distance = √[(1 - 1)² + (3\sqrt{3} - 0)²]
= √[0 + (3\sqrt{3})²]
= √[27]
= 3√3

However, we need to consider the full diagonal across the hexagon, which spans two sides. The diagonal $DA$ actually represents the vertical distance from the bottom vertex to the top vertex, which is twice the height of one of the equilateral triangles formed by the hexagon.

The height of one equilateral triangle with side length 10 is given by:

Height = (√3/2) * side length
= (√3/2) * 10
= 5√3

Since diagonal $DA$ spans the height of two such triangles, we multiply the height by 2:

Length of diagonal $DA$ = 2 * 5√3 = 10√3.

Thus, the final answer is $10\sqrt{3}$.<|END_OF_LATENT|> The final answer is $10\sqrt{3}$.