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Slope  =  [ 2  - - 7 ]  / [ -8 -4 ]  =   9  /  -12   =  - 3/4

Using (4, -7)

y = - (3/4) ( x -4)  - 7                    multiply through by 4

4y  = -3 ( x -4) - 28

4y  =  -3x + 12   - 28

3x + 4y  =   - 16

        P                    Q

                    4

                   X

                   8

   S                               R

PXQ  and RXS are similar triangles

The scale factor from  RXS and PXQ =  sqrt (8/4)  = sqrt (2)    /  1

Thus the height  of RXS =  sqrt (2) h / ( 1 + sqrt 2)              where h is the height of the trapezoid

And the height of PXQ  =  h /( (1 + sqrt 2) 

And  the base of PXQ =   base of RXS/sqrt 2

Using PXQ

Area =  (1/2) (base of RXS)  /sqrt (2) * h / ( 1 + sqrt 2)

4 = (1/2) (base RXS) * h / ( 2 +sqrt 2)

8(2 + sqrt 2) /  base RXS  =  h

Area of trapezoid =  (1/2) h (sum of bases)

Area of trapezoid =  (1/2)[ 8 ( 2 + sqrt 2) ] / base of RXS ]  [ base of RXS + base PXQ   ]  =

 4 (2 + sqrt 2)  / [ base of RXS ]  * [ base of RXS * ( 1 + 1/ sqrt 2) ]  = 

 4 [ 2 + sqrt 2 ] [ 2 + sqrt 2 ]  / / 2    =

2 [ 2 + sqrt 2] ^2  =

2 [ 4 + 4sqrt 2 + 2] = 

2 [ 6 + 4sqrt 2]  =

12 + 8sqrt 2

      X                Y

90

W60                          30  Z

Triangle ZXW  is a 30 - 60 -90  right triangle

Angle XZW  = 30°

3x - 6y  = 12 + 8x -3y            simplify

-3y = 5x + 12                      divide through by  -3      

y = (-5/3)x - 4

y  intercercept = (0, -4)

x intercept

0 = (-5/3)x  -4

4 = (-5/3)x

x = -12/5 = -2.4

Distance between   intercepts  =  sqrt [ (-2.4)^2 + (-4)^2 ] =  sqrt [ 544/25 ] =  (4 sqrt 34  )  /  5

Points are  (0,14)  and (x , 0)

slope =  ( 14 - 0 )  / ( 0 - x)  = 8

14 / -x = 8

-x = 14/8  = -7/4   = x intercept

Radius of large circle =  9......area =  pi * 9^2  =  81 pi cm^2

Total area of smaller circles=  (7) pi (3)^2  = 63 pi cm^2

Total area of a's  =  81 pi - 63pi  =   18pi cm^2

Using the square of the  distance  formula

(a-2)^2  + (2a - 7  - -1)^2 =  5

(a -2)^2  +  (2a -6)^2  = 5

a^2 -4a + 4 + 4a^2 -24a + 36  =  5

5a^2 - 28a + 35  =   0

Sum of possile a's  =  28  / 5

We set the smaller base to x, so the area of the trapezoid is

\(12x+24 = 80\)

\(12x = 56\)

\(x = \frac{19}{3}\)

Next, we find the diagonals by using pythagorean theorum.

a = 2

b = 12

c = \(\sqrt{148}\) = \(2\sqrt{39}\)

so the perimeter is 2x+4+2c = 38/3 + 4 + 4 root 39, or \(\frac{50}{3}+4\sqrt{39}\)

\(2x^2 - 8x + 7 = 5x^2 - 13x + 1\)

\(3x^2 - 5x -6=0\)

\(a = \frac{5+\sqrt{97}}{6} ,b = \frac{5-\sqrt{97}}{6}\)

You can take it from here.

 Hope this helped,

 IC

AGAIN......need the image.....    SMH 

a) expected value   =   (8 x 3 +  2 x 8 ) / 10 = 4

b)                                  (8 * 3  + 3 * 8 ) / 11 = 

c)                                   ( 8 * 3  + 4 * 8) / 12 = 

 d )                                    (8* 3 + x * 8) / (8+x)  = 6       solve for x

   e )                                  ( 8*3  + x*8) / (8+x) = 7         solve for x 

Need image.....

Here is what I would do: area of red rectangle =   5 x 3 = 15  units^2 

   then subtract each of the right triangles    area = 1/2 b * h 

        15 - 1 - 2 - 1/2  - 2.25  =   9.25 units^2 

Let a and b be the solutions to \(5x^2 - 11x + 4 = 4x^2 + 11x + 13\). Find \(\frac{1}{a^3} + \frac{1}{b^3}\).

For this question, I'm just going to solve for a and b, since it is not that hard.

You can use the quadratic formula, which is \(\frac{-b+or-\sqrt{b^2-4ac}}{2a}\). The +or- just stands for plus or minus. Any moderators tell me what the symbol for that is? I kind of forgot. Anyways, by plugging in the formula to the equation, first we got to simplify it.

\(5x^2 - 11x + 4 = 4x^2 + 11x + 13\)

\(x^2 - 22x -9 =0\)

\(x = \sqrt{130}+11,-\sqrt{130}+11\)

You can take it from here.

PS: I think you made a mistake, this answer looks kinda weird. Maybe it wasn't +11x it was -11x, so it would be a perfect square and easier to solve.

Well, to make it divisible by 8, the first 3 digits have to be divisible by 8. Since we know it is a palindrome, it is in the format \(abba\). We also know that \(bba\) is divisible by 8. The total number of these are 008, 112, 224, 336, 440, 448, 552, 664, 776, 880, 888, 992, which is 12 total posssible combinations.

Hoped this helped, IC

Please include a picture.

All the sides of a triangle have integer length. The perimeter of the triangle is 5 and the triangle is isosceles. How many such non-congruent triangles are there?

Well, we know that 2 sides of the triangle are the same, since it is iscoceles. that means that the 2 sides can either be 2 or 1. But, we can rule out 1, since 2 sides of a triangle musts add up to be more than the 3rd side, but if the iscoceles sides are each 1, then 1+1<3, so that is not true. That leaves only one option, which is that iscoceles sides are 2, and the other side is 1. 

y = 1 (x) 

find perpendicular line 

y = -x 

that passes through the point (5,7) 

y = -x + 12 

Intersection point is 6,6     ( found by equating    y =  x   and   y = -x +12 ) 

(6,6) IS THE midpoint BETWEEN the original point (5,7) and the reflected point (x,y)  

    so      ( 5+x ) / 2 = 6    and    (7+y) / 2 = 6    shows  (x,y) to be    (7, 5) 

This is pretty easy to see with a line  y = x .....but this is the method to use for more diffiicult line equatins such as ' y = 5x '    etc......

There are 2 Americans ...they can be either 1st 2nd 3rd =   6 ways 

   the 5 Canadians fill one of the two remaining spots = 10 ways                  10 x 6 = 60 ways for ONE cnadian and ONE American to medal together 

        the final spot is then for one of the remaining 12 

                12 x 60 = 720 ways 

If a Canadian or American can win the third spot , there are 17 skaters that can fill the third spot 

               17 x 60 = 1020 ways 

(I think) 

x^2 - 10x + y^2 + 10y + c = 0      Arange to standard circle form of an equation

x^2 -10x  + 25      + y^2 +10y + 25   =  -c  + 25 + 25 

(x-5)^2      +   (y+5)^2   = - c +50               circle centered at  5, -5    with radius^2 =  - c + 50 

                                                                                      if radius = 1    radius^1 = 1       then     - c + 50 = 1        c = 49 

First, simplify to :

3x^2 -15x - 90      then manipulate ot vertex form 

3 ( x^2 - 5x)   - 90 

3 ( x- 2.5)^2    - 18.75 - 90 

3 (x-2.5)^2 - 108.75                            <===== shows vertex at   2.5 , -108.75        n = - 108.75