there is a hexagon of vertices A,B,C,D,E,F inscribed in a circle of unit radius
what will be the product of line segments AB,AC,AE
I've done a quick drawing in paint just to illustrate my solution.. Its a pretty crude drawing but hopefully you can understand it!
The black line is the circle, red line is the hexagon and blue line joins opposite points.
I'm ASSUMING that this is a REGULAR hexagon. I.e. All sides are same length, same internal angles.
You'll notice that a hexagon is made up of 6 equilateral triangles as in my drawing.
This means that one side of the hexagon is equal to one side of the triangle.
Looking at my drawing you can see that 2 sides of the triangles form the radius of the circle.
So the side length of one triangle is just the radius divided by 2 which is 1 in this case.
Since every side of the triangle is the same length. The edges of the hexagon have length 1 as well. I.e. AB has length 1.
Now notice that AC and AE have the same length.
The direct distance from A to E is just twice the HEIGHT (i.e. distance from middle of base to opposite point) of one of the triangles.
This is found using Pythagoras rule $\displaystyle a^2 + b^2 = c^2$ with $\displaystyle c$ being the hypotenuse (longest side) of the triangle.
So in this case we try to find $\displaystyle a$. $\displaystyle c$ is the length of one side of the triangle so we have $\displaystyle c=1$. $\displaystyle b$ is the length from one of the points to midway along a joining edge which is just 0.5.
So putting these into the formula you get $\displaystyle a^2 + 0.5^2 = 1^2$ i.e. $\displaystyle a^2 = 1^2 - 0.5^2 = 1 - 0.25 = 0.75$. So $\displaystyle a = \sqrt{0.75}$. Hence AE = $\displaystyle 2 \cdot \sqrt{0.75} = \sqrt{3}$ = AC!
My god so many mistakes in this post...
This attachment is just to show you the a,b,c thing we did...
Look at the green triangle. c is the longest side. b the shortest, and a the one from one the point to the middle of the base that we want to find out.
The dotted line is to show we just have to double a to get our answer.
Refer to Deadstar's image for clarity.
Radius is 1.
The equilateral triangle makes distance AB = 1
Distance AD (the diameter) = 2
Distance CD (same as AB) = 1
Angle DCA or Angle ACD is 90 degrees.
The hypotenuse is [AD] is 2; the shorter side [DC] is 1.
Distance AC = $\displaystyle \sqrt{ 2^2 - 1^2 } =\sqrt{3}$
Line segment AE is the same length as line segment AC
The product of the line segments AB x AC x AE :
$\displaystyle 1 \cdot \sqrt{3} \cdot \sqrt{3} = $
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