CPhill

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 #1
avatar+81072 
+1

We have

 

   x  =       1 +   √2

                        _________

                        1 + √2

                              _________

                                1  +    .....

 

 

We can write

 

x  =   1  +  √2

                ___

                   x                    multiply both sides by x

 

 

x^2  =  x  +  √2

 

x^2  -  x   - √2   =  0

 

The solutions to this   are   

 x  =    1 /2 + √ [ 1  + 4√2]  / 2      and x  =     1/2 - √ [ 1  + 4√2]  / 2 

 

Evaluating      1 /  [ ( x  + 1) (x - 2) ]   for either value of x gives

 

Here's the  detail when  x  = the first value...you can check that the other value gives exactly the same thing  for   l A l  + l B l  +  l C l

 

                                  1

       _________________________________________________

       (  1 /2 + √ [ 1  + 4√2]  / 2   +  1)  (  1 /2 + √ [ 1  + 4√2]  / 2  - 2)

 

 

                                    1

      __________________________________________________

         (  √ [ 1  + 4√2]  / 2   + 3/2 ) (   √ [ 1  + 4√2]  / 2   - 3/2 )

 

 

                                   1

                 ______________________

                   ( [ 1  + 4√2]  / 4  -  9/4 )

 

 

                                    1

                     _________________

                              √2  -   2

 

 

                               2  +  √2

                             ________

                                  -2     

 

 

 

 

 

So  ....  A, B=  2 and C  =  -2    and    l A l  +  l B l  + l C l  =   6

 

 

cool cool cool

CPhill 1 hour ago
 #1
avatar+81072 
+1

1) Find the area of a regular 12-gon inscribed in a unit circle.

 

We have 12 identical triangles  identical isosceles triangles with equal sides of 1 and whose apex angle  between these sides =  360/12  = 30”  

 

The  area will  be given  by    (1/2) (1)^2*sin (360/12°)  =

 

(1/2 sin (30°)  =  1/2  *  1/2  =   (1/4 ) units^2

 

 

2) A regular hexagon has side length 6. If the perimeter and area of the hexagon are p and A, respectively, what is the value of (p^4)/(a^2)?

 

The perimeter, p, is 36  ⇒   p^4  =   36^4

The area, a,  is   (1/2)6^2sin (60)  =  18*√3/2  =  9√2  ⇒ a^2  = (√182) ^2   = 162

 

So p^4 / a^2  =    36^4  / 162   =   10368 

 

 

3)Isosceles triangle OPQ has legs OP = OQ, base PQ = 2, and and angle POQ = 45 degrees. Find the distance from O to PQ.

 

 

The distance from O to PQ  is the altitude....call the point where the altitude intersects the base, R

And this altitude bisects POQ....so angle POR  =  22.5°

And the altitude also bisects the base.... so PR  =  1

 

Using the tangent function....we have that

 

tan (22.5)  =  1 / altitude        rearrange as

 

altitude  =   1  / tan (22.5)  ≈ 2.4142  =    1 + √2   

 

 

 

4) A, B, C, D and E are points on a circle of radius 2 in counterclockwise order. We know AB = BC = DE = 2 and CD = EA Find [ABCDE].
Enter your answer in the form x + y√z in simplest radical form.

 

 

This is a regular pentagon inscribed in a circle....I assume you want the area of ABCDE??

If so...... the area is   5 (1/2)(2)^2sin (72)  = 10sin (72)  = 10 √  [  5/8  + √5/8 ]  units^2

 

If you want the perimeter...we  have that  the half side lengh  = 

2sin(36)

And we have 10 half side lengths comprising the perimeter....so....the perimeter  =

10 * 2 *  sin(36)   = 20√ [ 5/8  - √5/8 ] units

 

 

cool cool cool

CPhill 21.01.2018 17:25