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The side of a gray square is √2  and its area is 2 square units

The area of a blue square         A = 1/2[(√8 -2)²] = 0.34314575

The difference between these 2 areas is      1.65685425 square units

We need the cost of one gallon of gas to solve this......

      Sum >  Product

         1        2       3       4        5

1       >       >        >       >        >       

2       >      =        <       <        <

3       >      <        <       <       <

4       >      <       <        <       <

5       >      <       <        <       <

P (sum < = product)  =  16/25

THX, JoLink

Here's a slightly  easier way

[ (1)(14)  + (2)(5)  + (3)(9)  + (4)(7)  + (5)(7)  + (6)(8)  ]

_____________________________________________  =

       14   +  5     +    9      +   7    +   7    +   8

[  14  + 10  + 27  +   28  +  35   +  48 ] 

_______________________________   =

                   50

162   / 50     ≈     3.24.......corrected   !!!!

m/n + n/m    =  

[ m^2  + n^2 ]   /   mn

Now  just fill in what we  know

[ 8 ]  /  [ 4  ]  =   2  =  m + n

Triangles AFC  and  GHC  are similar

So  AF /FC  = GH/HC

AF/ GH  = FC /HC

6/4 =  FC /HC

3/2  = FC /HC

3/2= 9/HC

(3/2)HC  =FC

(3/2) HC  = 9

HC  = 6

So FH  = 3

And triangle GFH is similar to triangle DFC

So

FC/ FH  = DC/ GH

9/3 =  DC/ 4

3 = DC/ 4

DC  = 12

And we have  these relationships

BF/ AF = AF/ FC                               FC /DC = DC / CE

FC * BF = AF^2                                 CE * FC =  DC^2

9 BF  =  6^2                                       CE * 9 = 12^2                                        

BF  = 36/9  = 4                                  CE  =144/9   =  16

Area of the polygon is  =  area of ABC  +  area of FDE  - area of FGC  =

(1/2)(BF + FC) (AF)  +  (1/2) (FC + CE) (DC)  - (1/2) (FC)(GH)   =

(1/2) (4 + 9)(6) + (1/2) (9 + 16)(12)   -  (1/2)(9)(4)  =

       39       +  150    -   18  =

171 units^2

ok thanq! TY :D

Sorry i did it the long hard and painful way.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6

The Average is 3.24

Two parallel chords in a circle have lengths 6 and 8. The distance between them is 1. Find the diameter of the circle.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AB = 6         CD = 8         MN = 1

3² + (NO + 1)² = 4² + NO²          NO = 3

Diamater     D = 2* sqrt(NO² + 4²) = 10 units

We can form two right triangles

The longer chord is  closer to the center

So....if we  bisect both chords with a radii drawn from the center  we  have  that

r^2 =  sqrt  [ x^2  + 4^2]     where x is the  distance  from  the  center to the bisection point

Also......we  can  

r^2  = sqrt  [ ( x + 1)^2   + 3^ 2]

Set  these equal

sqrt  [ ( x^2 + 4^2 ]   =  sqrt  [ (x + 1)^2  + 3^2 ]    square  both sides  and simplify

x^2 + 16  = x^2  + 2x + 1  + 9

2x  = 6

x  = 3

The  radius =  sqrt  (3^2  + 16] =  sqrt (25)  =  5

The diameter = 2*5  = 10

                         x^2   +  3x  +    3

x^2  -3x  +  6  [ x^4  +  0x^3   +  0x^2  +   0x    +   1    ]

                        x^4  -  3x^3     +  6x^2

                      ________________________

                                  3x^3    -  6x^2   +  0x

                                  3x^3     - 9x^2  + 18x

                                ______________________

                                                 3x^2  - 18x  +  1

                                                 3x^2  - 9x    + 18

                                               __________________

                                                          -9x    -  17   =  the remainder

In the diagram shown here (which is not drawn to scale), suppose that triangles ABC and PAQ are similar, and triangles ABQ and QCP are similar.  If angle BAC = 60 degrees, then compute angle PQC.

We can use any type of triangle including a right-angled triangle.

All triangles are right-angled triangles, 30-60-90.

Angle PQC = 60 degrees

The Answer is 3!

Thank You!

The easiest way to start is to multiply both sides by 30   (that will get rid of all the fractions.)

Hello There,
there are 12 combinations.

Let f(x)=2x-3 For what value of x is (f(f(x)) = x?

\(f(x)=2x-3\\ so\\ f(f(x))=2(2x-3)-3\\ \text{so when does}\\ 2(2x-3)-3=x \)

well you will have to solve it to find out ;)

(7 x 11) + (7 + 11) = 95

7 + 11 = 18

The ratio is 9 / 100

Lets lay out these points in their appropriate quadrants.

(-5, 4)                         (5, 4)

(-5, -4)                        (5, -4)

If we look at the horizontal direction, the x coordinates in the boundaries are -4, -3, -2, -1, 0, 1, 2, 3, 4

If we look at the vertical direction, the y coordinate in the boundaries are -3, -2, -1, 0, 1, 2, 3

If you draw them out, you will see.

The horizontal has 9 x-coordinates.  The vertical has 7 y-coordinates. These points form an array inside the rectangle. If we multiply 9 by 7, we get 63.

There are 63 integer coordinate points inside this rectangle.

Triangle ABC has sides AB = 6, AC = 8, and BC = 12. The golden rectangle is inscribed in a triangle so that two of its vertices are on side BC and the other two vertices are on sides AB and AC. Find the area of a rectangle. The longer side of a rectangle should be on the side BC.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : (1 + √5) / 2

If the short side of a rectangle is x then the longer side is x * [(1 + √5) / 2] 

Find angles B and C using the law of cosines.

Use this formula to calculate the sides of a rectangle: 

                  x / tan∠B + x * [(1 + √5) / 2] + x / tan∠C = 12

These poitns with integer coordinates  are known as lattice points

Let A  = (5,4)   B = (-5,4)  C  = (-5, -8)   D  = (5, -8)

AB =  10      BC = 12

The  number  = (AB  -1)  ( BC  -1)   =   (10 - 1)  ( 12 -1)  =   9 * 11   =   99

(1/2) y + (1/4)  = (1/12) ( 3 + ( ) y)

(1/2)y  + (1/4)   = 3/12  + ( ) y / 12

(1/2)y  + 1/4  =1/4  +  ( ) y  /12       subtract 1/4 from both sides

(1/2)y =  (6 /12) y

6  wil produce infinite solutions 

Side of grey square  = sqrt (2) units

Area  of grey square=  (sqrt (2) )^2 =  2

 Distance  between  circle drawn  through the diagonal of the  square =  sqrt (8)

Diagonal of  blue square =   sqrt (8)  - 2

Side of  blue  square =   [ sqrt (8)  - 2 ]  /sqrt (2)

Area of blue square = (1/2) (sqrt (8)  -2)^2   = .343

Difference  of areas  =   2 - .343 ≈  1.657  units^2

Perimeter  =  124

2(BC + CD)  = 124

BC + CD  =  62

CD  =  62 - BC

Area  of Paralleogram   = Area of Parallelogram

BC ( AE)  =  AF ( CD)

BC ( 7)  =  24 ( 62  - BC)

7BC  = 1488  - 24BC

31 BC =  1488

BC  = 1488/31  =  48

So....the area =  BC * 7 =  48 * 7   =   336 units^2

It's 12, because triangle ABC is larger than triangle BDC by a scale factor of 2. The triangles are similar. Therefore, AC is corresponding to BC. AC is 8 units, are AD is 8 - 2 = 6 units. AB is corresponding to BD, and 3 * 2 = 6 units. 6 + 6 = 12 units.