CPhill

Simplify as

4x^2 - 8x + 4y^2 + 14y  =  -12          divide through by 4

x^2 - 2x + y^2 + (7/2)y  =  -3            complete the square on x, y

x^2 -2x + 1  + y^2 + (7/2)y + 49/16  = -3 + 1 + 49/16

r^2  = -2 + 49/16  =  [- 32 + 49 ] / 16  =  17 /16

r  = sqrt (17) /  4

Slope of the given line  = 1

Slope of the perpendicular line through A  = -1

Equation of this  line

y = -1 (x -7) + 1

y = -x + 8

x coordinate of the intersection of these lines

x + 3 = -x + 8

2x= 5

x  = 5/2  = 2.5

y = (2.5) + 3 =  5.5

B  = ( 2.5 + (2.5 - 7)  ,  5.5 + (5.5 -  1) )  =  (-2 , 10)

        Math                Science

        Club                 Club 

       Only     Both     Only

        0            3             3

Math Club Only  + Science Club  Only   (but not both)  =

       0                  +      3  =                  3

ST =  (5/7) .....SR = (5/7)QP  →  SR / QP = 5/7 

Triangles SUT  and  QUP  are similar

So VR   = (5/7) QV  →  VR /QV  = 5/7

So  QV =  (7/12)QR  →  QV / QR  = 7/12

UV is parallel to SR

So 

QV /QR  =UV /SR

7 /12 =  UV / 20

UV = (20)(7/12)  =  140/12  = 35 / 3 =  11 + 2/3

1.  5cos A + 12sinA + 12        take the derivative and set to 0

-5sinA  + 12 cos A  =  0

12cos A = 5sinA

sinA / cos A  = 12/5

tan A  = 12/5

cos A =  5/13       sin A =  12/13

The max value is

5 (5/13)  + 12 (12/13) + 12   =   (25 + 144) / 13 + 12  =  169/13 + 12  =  13 + 12 =  25

cos PMR  = -cos PMQ

Law of Cosines (twice)

PO^2  = QM^2  + PM^2 - 2(QM * PM) cos PMQ

PR^2  = RM^2 + PM^2 - 2(RM * PM) ) (-cos PMQ)

18^2  = 12.5^2 + PM^2 - 2(12.5 * PM) cos PMQ

23^2 = 12.5^2 + PM^2 + 2(12.5 * PM) cos PMQ     add these

18^2 + 23^2  =  2*12.5^2 + 2PM^2

853 = 312.5 + 2PM^2

540.5 = 2PM^2

270.25 = PM^2

sqrt [270.25] = PM ≈ 16.44

Number of diaginals in a regular polygon =   (n) (n-3) / 2   ....  where n = the number of sides

So

(n) ( n -3)  / 2  =  2n

n^2 - 3n  = 4n

n^2 - 7n = 0

n ( n  - 7) = 0

n - 7 = 0

n = 7 

A = 6 (1/2) (side length)^2 * sqrt (3) / 2

3 = 3  (side length)^2 * sqrt (3) /2

2 /sqrt 3  = side length^2

sqrt [ 2 / sqrt 3 ]   = side lengrh ≈  1.075

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