CPhill

1) In concave equilateral pentagon ABCDE, Angle A is qual to Angle B is equal to 90 degrees. What is the degree measure of angle E?

I believe we have something like this  :

Angle A  =  B   = 90°   and the pentagon is equilateral

If we connected  EC, triangle  DEC is equilateral.....so angle DEC  = 60°

And angle AEC  =  90°

And angle AED  =  AEC - DEC  =  90°  -  60°   =  30°  =  angle  E

1/x  +  1/y   =   1/7

[ x + y ] /  [xy]  =  1/7

7x +  7y   =  xy

7y -  xy   = -  7x

y [ 7  -  x ]  =  - 7x

y  =   [  7x ]  /  [ x -  7  ]

Note that  as  x ⇒  ±inf,  y ⇒  7      

y will  be an integer  when

x  =  -42    y   =   6   

{no need to  test  any  x  values less than this since y will be < 7}

x  =   6      y  =  - 42

x =   8       y  =    56

x  =  14     y  =   14

x =  56      y  =   8 

{no need to test any x values more than this since y will be > 7}

Thanks, EP.........

183144246353322528363047                    113  

__________________________    =          ___ 

999999999999999999999999                    617

Thanks, EP.....the parent function  is   y  = x

So.....x + y  = 3  ⇒     y  =  -x  +  3    

So.......this graph  reflects the parent function  across the x axis and  shifts it up 3 units

f(x)  =  [ x^2  -  25 ] / [ x + 5 ]  =  [ (x + 5) ( x- 5)]  / [ x + 5 ]  =  x - 5

This will be the graph of  f(x) =   x - 5  with a  "hole"  at   x  =  - 5

So.....the last graph is correct

With a dilation center at the origin, we  can  find the scale factor from the ratio  of  any image  coordinate to its pre- image coordinate

So.....using the x coordinate  of  A'  and A   we have

-3.75 / - 5    =    3.75 / 5  =    [ 3 * 1.25 ]  / [ 4 * 1.25 ]   =    3/4  =  scale factor

( x^2+y^2-z^2)^2 - 4x^2y^2

 ( [ (x^2 + y^2  - z^2 ]  +  2xy )   ( [ x^2 + y^2 -z^2] - 2xy )

(  [ x^2 + 2xy + y^2 ]  - z^2  )  (  [ x^2 - 2xy + y^2]  - z^2 )

(  ( x + y)^2  - z^2 )   (  ( x - y)^2  - z^2 )

(  (x + y) + z )  (  (x + y ) - z )  ( (x - y) + z)  ( ( x - y) - z )

( x + y + z )  ( x + y - z) ( x - y  + z )   ( x - y - z )

The area of the square is, of course, 1 square unit

Each of  equilateral triangles has a side of 1, so  their total area  is

4  (1/2) * 1^2  * sin (60)  =    2*√3/2  =  √3  square units

The other 4 triangles have two sides   of 1  and  an included angle between these sides  of 

[360 -  90 - 60 - 60  ]  =  150°

So...their area is

4 (1/2) * 1^2 * sin (150)  =  2 * (1/2)  =  1 square unit

So...the area  of  the larger square  is   ( 1  + 1 + √3)  =  [  2 + √3 ]  units^2

B)   The probability that the first card drawn is a not square number  is  4/6

And the probability that the second card drawn is a not square number [ given that the first was not a square number]  is 3/5

So......the probability  that at least one of the cards  is a square number is  

1 - P(that neither is a square number)  =  

1 - (4/6)(3/5)   =

1 - 12/30  =

1 - 2/5  =

3/5   =  60%

The result is different from part A......here, we have  a dependent event......the outcome of the second card drawn depends upon the frist card drawn....however.....the number appearing on one die is independent of that appearing on the other

2) The measures of the angles a 9-gon are in ratio 3 : 1 : 4 : 1 : 5 : 9 : 2 : 6 : 5. What is the number of degrees in the measure of the largest angle?

The sum of the interior angles of a 9-gon   =  (9 - 2) * 180  =  7 * 180  = 1260°

So....the largest angle is given by

9 / [ 3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 ]   *  1260  =

9/ 36  * 1260   =

(1/4) * 1260   = 

315°