geno3141

By the Pythagorean Theorem, triangle(DEA) is a right triangle,

The area of triangle(DEA)  =  ½·5·12  =  30

Find the point F, on DA, that is directly across from point E.

1)  DCEF is a rectangle with DE dividing it into two congruent triangles. 

     Therefore, the area of triangle(DCE) = area of triangle(FED).

2)  ABEF is a rectaongle with AE dividing it into two contruent triangles.

     Therefore, the area of triangle(BAE) = area of triangle(FEA).

3)  Since the area of triangle(DEF) + area of triangle(AEF) = 30,

      the area of triangle(DCE) + area of triangle(ABE) = 30,

      and the total area of the rectangle is 60.

Since the number 5 is directly opposite the number 14, the numbers 6, 7, 8, 9, 10, 11, 12, and 13 must be between them.

Since there are 8 numbers between 5 and 14, continuing around the circle, there must also be 8 numbers between 14 and 5.

Four of them are 1, 2, 3, and 4; the others must be 15, 16, 17, and 18.

n = 18.

Assuming that the blue triangle is a right triangle:

  --  using the Pythagorean Theorem, we get the missing side to be 5

  --  the area will be:  ½·5·12  =  30 

The yellow triangle is a 30-60-90 triangle, so:

  --  the side opposite the 30^o angle is  13/2

  --  the side adjacent to the 30^o angle is  (13/2)·sqrt(3)

  --  the area will be:  ½·( 13/2 )·[ (13/2)·sqrt(3) ]  =  (169/8)·sqrt(3)

The difference in area is:  | 30  -  (169/8)·sqrt(3) |

27 · 9^3x  =  3^k

Rewriting with every factor having a base of 3:

3³ · (3²)^3x  =  3^k

3³ · 3^2 · 3x  =  3^k

3³ · 3^6x  =  3^k

3^{3 + 6x}  =  3^k 

k  =  3 + 6x

By the Pythagorean Theorem, we know that  x² + h²  =  a²   and   y² + h²  =  b²

which, if we add them together, we get:    x² + h²  +  y² + h²  =  a² + b²

However, we want to show that (x + h)² + (y + h)²  =  (a + b)²

and, this is:      ( x² + 2xh + h² ) + ( y² + 2yh + h² ) =  a² + 2ab + b²

So, somehow we will need to show tha 2xh + 2yh  =  2ab

We can do this by finding the area of the triangle in two different ways:

    Area  =  ½ab   and the area is also   Area  =  ½(x +y)h

So:  ½(x +y)h  =  ½ab

--->     (x + y)h  =  ab

--->     xh + yh  =  ab

--->   2xh + 2yh  =  2ab

Adding this to:  x²           + h²  +  y²           + h²  =  a²          + b²

we get:             x² + 2xh + h²  +  y² + 2yh + h²  =  a² + 2ab + b²

which is:              (x + h)²        +     (y + h)²        =      (a + b)²

Flip each of the equations:

1)  (xy) / (x + y)  =  1     --->     (x + y) / (xy)  =  1

2)  (xz) / (x + z)  =  2     --->     (x + z) / (xz)  =  1/2   

3)  (yz) / (y + z)  =  3     --->     (y + z) / (yz)  =  1/3

Rewrite each equation:

1)  (x + y) / (xy)  =  1        --->     x / (xy) + y / (xy)  =  1        --->    1 / y + 1 / x  =  1

2)  (x + z) / (xz)  =  1/2     --->     x / (xz) + z / (xz)  =  1/2     --->     1 / z + 1 / x  =  1/2

3)  (y + z) / (yz)  =  1/3     --->     y / (yz) + z / (yz)  =  1/3     --->     1 / z + 1 / y  =  1/3

Combining  1)  and  2):     1 / y + 1 / x  =  1

    subtract:                        1 / z + 1 / x  =  1/2

                                          1 / y - 1 / z  =  1/2                      (equation 4)

Combining  4)  and  3):     1 / y  - 1 / z  =  1/2

    add:                               1 / z + 1 / y  =  1/3

                                                    2 / y  =  5/6

    solving:                                        y  =  12/5         

Now, substitute this back into equation  1)  to find x  and into equation  3)  to find z.                                       

You subtract the radians that are inside the parentheses and you place that answer within the parentheses of

the answer..

Specifically:  a·cis( 7pi/6)  /  b·cis( pi/4)  =  c·cis( 11pi/12 )

You are given the value of a:  12sqrt(2)  and the value of b:  3sqrt(3).

You need to find the value of c by dividing the value of a by the value of b 

and you place this answer in front of the  "cis( )"  part  --  which is the empty box before (and, in this case, on the

line above "cis(    )"

If three angles contain a total of 120^o and the total of all the other angles is 150^o  --  and if we are talking about

interior angles  --  the sum of the interior angles is 270^o.

This is impossible ...   

I agree with the approach of the other answer, except that there are 540^o in a pentagon.

540^o - 180^o  =  360^o

360^o / 3  =  120^o

Try     (2 + 3i)(5 - 8i)