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8)In rectangle ABCD, E is the midpoint of BC and F is the midpoint of CD. Let G be the intersection of AE and BF. We know AB = 10 and BC = 15. Find DG.

Let  A  = (0,0)   B = (0, 10)  C  = (15,10)  D  =(15,0)  E  = ( 7.5, 10)  F = (15, 5)

The equation of the the line containing segment AE  is

y = (4/3)x

The slope of segment BF   is  [5 - 10] / [15 - 0] = -5/15  = -1/3

And the equation of the line containing BE  is

y = (-1/3)x + 10

So....the x coordinate of the intersection  of these two lines can be found as

(4/3)x  = (-1/3)x + 10

(4/3 + 1/3) x  = 10

(5/3)x  =10

x = 6

And the y coordinate of the intersection of these lines is

y = (4/3)(6)  = 8

So....G  = ( 6,8)

So  DG   =  √ [ (15 - 6)^2 + (8 - 0)^2 ] = √[9^2 + 8^2 ] = √[81 + 64] = √145 units

6)Given  A = (1,1), B = (4,2), C = (3,4)  and D = (2,3). Find the intersection of the diagonals of quadrilateral ABCD. Enter your answer in the form "(x,y)".

Probably several ways to do this, but let's get the equation of two lines and find their intersection

The line containing AC  has a slope  of   [ 4 - 1 ] / [ 3 - 1 ] = 3/2

And the equation of this line is

y = (3/2) ( x  - 1)  + 1

y = (3/2)x +=- 1/2

The line containing BD  has a slope  of [ 3 - 2] / [ 2 - 4]   = -1/2

And the equation of this line is

y = (-1/2)(x - 2) + 3

y = (-1/2)x + 4

To find the x intersection of these lines, we have

(3/2)x - 1/2  = (-1/2)x + 4

2x  =  9/2

x = 9/4

And the  y coordinate of the intersection is

y = (3/2)(9/4) - 1/2

y = 27/8 - 1/2

y= 27/8 - 4/8

y  = 23/8

So...the intersection of the diagonals  is  ( 9/4, 23/8)

5)The graph x^2 + y^2 = 100 intersects y = 3x at two points P and Q. Find PQ.

Sub the second function into the first function for y

x^2  +  ( 3x)^2   = 100

x^2  + 9x^2  =100

10x^2  = 100     divide both sides  

x^2  = 10       take both roots

x = √10     or    x  =  -√10

So....let P   =  (√10, 3√10)   and  Q  = ( -√10 ,  -3√10 )

So

PQ   =  √ [ (2√10)^2  + (6√10)^2  ]  =  √ ( 40 + 360)  = √400  =  20 units

3)I'm assuming that we have.... 3x - 4y =  24 intersects the x-axis and y-axis at points A and B respectively. Find distance AB.

x intercept  is  (8,0)   

y intercept  is  ( 0, -6)

The distance from A to B  is

√ [ ( 8 - 0)^2  + ( -6 - 0)^2  ]  =

√ [ 64  + 36 ] =

√100  =

10 units

2)Find the area of the region that is enclosed by the graph of the equation

4x^2 - 12x + 4y^2  + 56y -471 = 0      completre the square on x and y

4(x^2 - 3x  + 9/4)  +  4 (y + 14y + 49)  = 471  + 9  + 196

4 ( x - 3/2)^2 + 4 ( y + 7)^2  =  676

(x - 3/2)^2  + (y + 7)^2  =  169

This is a circle  centered  at  (3/2, -7)  with a radius of 13

The area   is      pi * r^2  =    169 pi units^2  

1)Find the equation whose graph is the line passing through the points (2, -3) and (4,5). Enter your answer in standard form Ax + By = C.

Slope   =   [ 5  -  -3  ]  /  [ 4 - 2 ]  =   8/ 2  = 4

Equation is

y  = 4 (x -4) + 5

y  = 4x - 16  + 5

y = 4x   - 11

4x  - 1y  =  11

Independent

P ( A l B)  = P (A)

P (two white cards)  = P(white on first draw) * P (white on second draw)

14 / 95   =   2 / 5   *   P (white on second draw)

Multiply both sides by 5/2

(5/2)(14/95)   = P (white on second draw)   = 70 / 190   =   7 / 19

P  (odd  then even )   =

(5/10)  (4/9)   = 

20 / 90 =

2 / 9 

We have 10 balls

P ( green  then  blue )  =   3 /10  * 2 / 10   =    6 / 100  =  3  / 50

The center  is  ( -9, -6)

We jhave 2 possible outcomes trhat do not show a "1"  on the first spinner

And there is only  1 outcome where the second spinner shows a "4"

So......  the number or outcomes  is  2 * 1   =  2

We have three possible outcomes  on every spin

So...on three spins, we have

3 spins  * 3 outcomes per spin  =  27  different outcomes  

y  = 18x^2 + 4x  + 20

y = 18  (x^2  + 4/18x   +  20/18)   complete the square on x

y  = 18  ( x^2 + 4/18x + 16/1296 + 20 / 18  - 16/1296 )

y  = 18 [ ( x + 4/36)^2 +  89/81]

y = 18(x + 4/36)^2  + 178/9

(y - 178/9 )  = 18 ( x + 4/36)^2

(1/18) (y - 178/9)  = (x + 1/9)^2

The vertex  is  ( -1/9, 178/9 )

This parabola opens upward

We  have that     4p  = 1/18    ⇒ p   = 1/72

So....the focus is  at    ( -1/9, 178/9 + 1/72)  = (-1/9, 475/24 )

The ladder length will form the hypotenuse of a right triangle and the 11.8 ft will form the opposite leg to the angle we are seeking....to find this angle, use the sine inverse....so we have

arcsin (11.8 / 12)  ≈  79.5°

So..it is unsafe

Here's a pic :

Thanks! 

x^2  + y^2  + 4x  + 2y  - 4   = 0

x^2 + 4x + 4  + y^2 + 2y  + 1  =  4 + 4 + 1

(x + 2)^2  +  (y + 1)^2   =  9

The center is  ( -2, -1)  and the radius is 3

EDIT TO CORRECT ASLIGHT ERROR  !!!

Ah, thanks. I do remember completing the square. 

We need to  complete the square on x and y for each one of these.....you should review how to do this if you are unsure of the procedure....I'll do the first one in some depth

1) x^2 + y^2 + 24x + 10y + 160 = 0

x^2 + 24x +  y^2 + 10y  = -160

Take 1/2 of 24, square it = 12^2  = 144...add to both sides

Take 1/2  of 10, square it  = 5^2  = 25....add to both sides

x^2 + 24x + 144  + y^2 + 10y + 25  = -160 + 144 + 25  factor and simplify

(x + 12)^2  + ( y + 5)^2  = 9

The center is  ( -12, -5)  and the radius  = 3

2) x^2 + y^2 - 6x + 10y = 97

x^2 - 6x + 9  + y^2 + 10y  + 25  =  97  + 9 + 25

(x - 3)^2  + ( y + 5)^2  = 131

The center is  ( 3, -5)  and the radius  is √131

3) x^2 + y^2 +26x + 28y = -364

x^2 + 26x + 169  + y^2 + 28y + 196  = -364 + 169  + 196

(x + 13)^2  + ( y + 14)^2  =  1

The center is  (-13, -14)  and the radius  is 1

If the difference in the largest and smallest angle is 44 degrees.....then  the common difference between each successive angle measure is 11°

And the sum of the angle measures  is 540°

Let n   be the smallest angle....so we have

n +  (n + 11) + (n + 22)  + (n + 33)  + (n + 44)   = 540     simplify

5n  + 110  =   540       subtract  110  from both sides

5n  = 430      divide both sides by 5

n  = 86° =   the smallest angle

27.  9y^2 - 16x^2   = 144

Divide through by 144 to get to this form

y^2   -  x^2     = 1

___     ____

16         9

The center  is  (0, 0)

The  foci  are given  by

(0, ±√ [ 4^2 + 3^2 ] =

(0, ±√25 )  =

(0, ±5)

28.  The correct equation is

y^2    -  x^2    

___      ___  =    1

144       25

1. Opposite angles in an inscribed quadrilateral are supplementary.....thus  angles  D  and F  sum to  180°

2. Correct  !!!

3. Don't know this one

4.Since DE is parallel to AC, DE splits the sides of triangle BAC proportionally.....this means that

DA/BD  = EC/BE

5. sin C  = h/a

   The other answers are correct  !!!

6. Definition of an altitude

psin M  = h  and msinP  = h

Substitution Property of Equality

sin M / m  = sin P / p

1. Correct  !!!

2. A radius always meets a tangent at right angles.....thus HJK is a right angle

3.   We have at riangle with a base of 4 and a height  of 5

So....its area   is   (1/2) (4) (5)   = (1/2) 20  = 10 units^2

Also.....we have a semicircle  centered at ( -1, -2) and a radius of 2

So...its area  is   (1/2) (3.14) (2)^2 = 2 * 3.14  = 6.28 units ^2

So...the total area  is  [ 10 + 6.28] = 16.28 uniits^2

4. Correct !!!

5. Correct  !!!

23)

y=x^2 -8x +18

y -18 = x^2-8x

y-18 +16  = (x-8x+16)

y-2 = (x-4)^2

vertex(4,2)

Because (x-4)^2 has a coefficient of 1, and (y-k)= 1/4p(x-h)^2, p's value is 1/4.

Because x has a positive coefficient, the parabola is pointing upwards, so the focus is above the vertex. 

Focus(4, 2.25)

Directrix( y = 1.75)

24).

(x-2)^2 + (y-1)^2 = 16 

becomes 

(x-2+4)^2 + (y+1-6)^2 = 16

cleaned up 

(x+2)^2 + (y-5)^2 = 16

20).

June =                                $22

July =              22*1.1 =       $24.2

August =        24.2*1.1=        $26.62

September = 26.62*1.1 =       $29.282

October = 29.282*1.1 =           $32.2102

November = 32.2102*1.1 =       $35.43122

December = 35.43122*1.1 =       $38.974342

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Total:                                            $208.717542 ≈ $208.72