CPhill

The point (3,4)  is  on the original graph

f(x)  + 4  shifts this up by 4 units

So   ( 3, 8)  must be on the graph

Midpoint BC  =  [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]

Midpoint AB  = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]   

Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]   

Slope  AD  =  [ 4 - 1 ] / [ 5 - -4] =  1/3

Slope CE  = [ -4-5] / [ 1 - -2] =  -3

Slope BF =  [ 6-0] / [ -9-3]  = -1/2

Equation of line  through AD

y = (1/3)(x + 4) + 1                   (1)

Equation of line  through CE

y =  -3 ( x + 2) + 5              (2)

Equation of line  through BF

y = (-1/2)(x -3)       (3)

Set (1), (2)  equal  to find their x intersection

(1/3)(x + 4) + 1 =  -3(x + 2) + 5

x + 4 +  3  =  -9(x + 2) + 15

x+ 7 = -9x -18 + 15

x + 7 = -9x -3

10x = -10

x = -1

y = -3(-1+2) + 5 =  2

Intersection pt  =  ( -1 , 2)

Test in (3)

2  = (-1/2) (-1 -3) 

2 = 2         true  

Total area =  DC * AD

Area of triangles  ADU + BCT  = 2* (1/2)( DC/3) (AD)  =  DC * AD  /  3

Shaded area  =  DC*AD - DC * AD / 3  =  (2/3)DC *AD  = (2/3)(180) =  120

Call the total number of games to be played  = T

We have played  .60T

We have  .40T  left to play

We have won  .40 (.60)T =  .24T

We need to win .50T

So

[  Games  we need to win - Games we have won ] / [ Games to be played ]   =  %we need to win

[ .50T  - .24T ] / .40T  =

.26 / .40   = 

13 / 20  = 

65%  of  our  remaining games

x^2 - 22x + c

c =  (22/2)^2  = (11)^2   =121

x^2 - 22x + 121  =   ( x - 11)^2

(1/2)BC = 4

Call the altitude  AD =   sqrt [ 5^2 - 4^2]  = 3

tan ( ABC)  = (3/4)

tan (ABC / 2)  =    1 -cos (ABC) / sin (ABC)  =  [ 1  - 4/5 ] / [(3/5 ] = 1/3

Equation  of angle bisector  of  ABC  =   y =(1/3)x

Intersection of   this line with  angle bisector BAC {line  x = 4 }

y= (1/3) (4)  = 4/3

Height of triangle  AMN  =  3 - 4/3  =  5/3

Area  of  [ABC]  =  (1/2) AD * BC  =   (1/2) (3) (8)  = 12

Triangle AMN  similar to triangle ABC

Scale factor  of  AMN/ ABC  =  (5/3) / 3 =  5/9

[ AMN ]  =  12 *  [ (5/9 ] ^2  = 100 / 27   ≈  3.7

A                     X       S        R          B

XB =  2808 / 2  =  1404

XR = (3/5)XB (3/5)(1404)  = 842.4

XS = (2/17)XR = (2/17)842.4 = 8424/  95

RS =  XR - XS  =  842.4 - 8424/95   = 71604/95 ≈ 753.73

(x + 2) / x  *  ( x + 3) / (x + 1)  *  ( x + 4)  / (x + 2)   = 10

(x + 3) / x * ( x + 4)  / ( x + 1)  =  10

(x + 3) ( x + 4)  = 10 (x) (x + 1)

x^2  + 7x + 12  = 10x^2 + 10x

9x^2 + 3x - 12  = 0

3x^2  + x  - 4  =  0

(3x + 4) ( x - 1)  = 0

(x - 1)  = 0

x = 1

First fraction  = (1 + 2) / 1  =  3 / 1

r =  .036 / .4   =  36 / 400  =  9 / 100 =  .09

Sum  =   first term / [ 1 - r ]  =   .4  / [ 1 - .09 ] =   .4 / .91  =  40 / 91

Slope  of PQ =   [ 4 - -5 ] / [ 5 - -1] =  9 / 6  =  3 / 2

y =(3/2) (x - 5)  + 4        multiply through by  2/3

(2/3)y = x - 5 + 8/3

(2/3)y  = x  -  7/3           rearrange

x - (2/3)y  =  7/3           multiply by 3/7

(3/7)x - (3/7)(2/3)y = 1

(3/7)x - (2/7)y  =1

x / ( 7/3)   y  / ( 7/2)  = 1