In triangle ABC, BC = 20 * sqrt(3) and angle C = 30 degrees. Let the perpendicular bisector of BC intersect BC and AC at D and E, respectively. Find the length of DE.
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We have that $D$ is the midpoint of $BC$, and that $CD = BC/2 = 20 \sqrt{3}/2 = 10 \sqrt{3}$.
[asy]
unitsize(3 cm);
pair A, B, C, D, E;
A = dir(133);
B = dir(193);
C = dir(-13);
D = (B + C)/2;
E = extension(A, C, D, D + rotate(90)*(B - C));
draw(A--B--C--cycle);
draw(D--E);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
[/asy]
Also, triangle $CED$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle, so $DE = CD/\sqrt{3} = 10 \sqrt{3}/\sqrt{3} = \boxed{10}$.
To find the length of DE, we can start by drawing a diagram to better understand the given information.
1. Draw a triangle ABC with angle C = 30 degrees.
2. Label BC as 20 * sqrt(3).
Now, let's proceed with finding the length of DE:
Step 1: Find the length of AC.
Since angle C is 30 degrees and we have the length of BC, we can use trigonometry to find the length of AC.
In a right-angled triangle, the side adjacent to an angle is equal to the hypotenuse multiplied by the cosine of that angle.
cos(C) = adjacent / hypotenuse
cos(30) = AC / BC
√3 / 2 = AC / (20 * √3)
√3 / 2 = AC / (20/√3)
√3 / 2 = AC * (√3 / 20)
√3 / 2 = AC * (√3 / 20)
AC = (√3 / 2) * (√3 / 20)
AC = 3/40
Step 2: Find the coordinates of the perpendicular bisector's intersection point (D) on BC.
Since the perpendicular bisector of BC intersects BC at D, we need to find the coordinates of D.
Since the length of AD is equal to the length of CD (perpendicular bisector of a line segment divides it into two equal parts), we can place D at the midpoint of BC.
D = (BC / 2, 0)
D = (20 * √3 / 2, 0)
D = (10 * √3, 0)
Step 3: Find the coordinates of the perpendicular bisector's intersection point (E) on AC.
To find the coordinates of E, we need to understand the geometry of the triangle.
Since we know the length of AC and C is a right angle, we can conclude that angle A is 60 degrees (sum of angles in a triangle).
Now, we can find the coordinates of E using trigonometry.
In a right-angled triangle, the side opposite to an angle is equal to the hypotenuse multiplied by the sine of that angle.
sin(A) = opposite / hypotenuse
sin(60) = AE / AC
√3 / 2 = AE / (3 / 40)
√3 / 2 = AE / (3/40)
√3 / 2 = AE * (40 / 3)
√3 / 2 = AE * (40 / 3)
AE = (√3 / 2) * (40 / 3)
AE = 20 * √3 / 3
Step 4: Find the distance between D and E.
Now that we have the coordinates of D and E, we can find the length of DE using the distance formula.
DE = √((x₂ - x₁)² + (y₂ - y₁)²)
DE = √((10 * √3 - 0)² + (20 * √3 / 3 - 0)²)
DE = √(300 - 0 + 200 - 0)
DE = √(500)
DE = √(25 * 20)
DE = 5√20
Therefore, the length of DE is 5√20.