geno3141

The area of the whole circle is  pi · 5²  =  25pi.

The yellow area is 20pi, so the portion of the circle colored yellow is  20pi / 25pi  or 4/5^ths.

The number of degrees in the yellow portion of the circle is (4/5)·360^o  =  288^o.

This leaves  360^o - 288^o  =  72^o  for the angle in the triangle.

Now, you can calculate how many degrees are in the other angle of the triangle.

angle(ABC)  =  ½ ( arc(CD) + arc(DE) + arc(EA) )

angle(CDE)  =  ½ ( arc(EA) + arc(AB) + arc(BC) ),

angle(ABC) + angle(CDE  =  ½ ( arc(CD) + arc(DE) + arc(EA) + arc(EA) + arc(AB) + arc(BC) )

But arc(EA) = 180^o  and  arc(AB) + arc(BC) + arc(CD) + arc(DE) = 180^o,

so:  angle(ABC) + angle(CDE  =  ½ ( 180^o + 180^o + 180^o )  =  the answer ...

Draw CA.

CA will be perpendicular to CDE (radius to tangent).

DB || CA (both perpendicular to EC).

Find point X on CA such that DX || BA.

XA = 4 (DBAX is a parallelogram).

CX = 1.

DX = 5.

Triangle(DCX) is a right triangle   --->   CX² + CD²  =  DX²   --->   1² + CD²  =  5²

   --->   CD  =  sqrt(24)

Triangle(DCA) is a right triangle   --->   DC² + CA²  =  DA²   --->   [sqrt(24)]² + 5²  =  DA²

   --->   DA  =  7

Now, you can find the perimeter of triangle(DBA).

If  6  is one root, then  (x - 6)  is a factor.     So:  (x - 6)(x + ?)  =  x² - 2x + a

When you add  ?  to  -6  you must get  -2,  so  ?  =  4

The factors are (x - 6)(x + 4)  =  x² - 2x + 24

x + y + xy  =  19   --->   y + xy  =  19 - x   --->   y(1 + x)  =  19 - x   

   --->   y  =  (19 - x) / (x + 1)          [Equation #1]

y + z + yz  =  29   --->   y + yz  =  29 - z   --->   y(1 + z)  =  29 - z

   --->   y  =  (29 - z) / (z + 1)          [Equation #2]

z + x + xz  =  23   --->   x + zx  =  23 - z   --->   x(1 + z)  =  23 - z

   --->   x  =  (23 - z) / (z + 1)          [Equation #3]

Substituting Equation #3 into Equations #1:     

   y  =  (19 - x) / (x + 1)   --->   y  =  [ 19 - (23 - z) / (z + 1) ] / [ (23 - z) / (z + 1)  + 1 ]

Multiplying numerator and denominator by (z + 1)

                                       --->   y  =  [ 19(z + 1) - (23 - z) ] / [ (23 - z) + 1(z + 1) ]

                                       --->   y  =  [ 19z + 19 - 23 + z ] / [ 23 - z + z + 1 ]

                                       --->   y  =  [ 20z - 4 ] / [ 24 ]

                                       --->   y  =  (5z - 1) / 6         [Equation #4]

Setting Equation #2 and Equation #4 equal to each other:

                                           (29 - z) / (z + 1)  =  (5z - 1) / 6 

Cross-multiplying:                          6(29 - z)  =  (5z - 1)(z + 1)

                                                       174 - 6z  =  5z² + 5z - z + 1

                                                                  0  =  z² + 2z - 35

                                                                  0  =  (z + 7)(z - 5)

So:  either  z = -7  or  z = 5        (the problem specifies that z must be positive   --->   z = 5)

Equation #2:  y  =  (29 - z) / (z + 1)   --->   y  =  (29 - z) / (5 + 1)   --->   y = 4

Equation #3:  x  =  (23 - z) / (z + 1)   --->   x  =  (23 - 5) / (5 + 1)   --->   x = 3  

There are two cases (Guest answered the first one):

Case 1:  z + 1  >=  0:     --->     3 - z  >=  (z + 1)·1

                                                 3 - z  > =  z + 1

                                                      2  >=  2z

                                                      1  >=  z     --->     z  <=  1     (-inf, 1]

 Case 2:  z + 1  <= 0:     --->     3 - z  <=  (z + 1)·1                    (change the direction of the inequality

                                                 3 - z  <=  z + 1                            because you are multiplying by a

                                                      2  <=  2z                                negative)

                                                      1  <=  z     --->     z  >= 1     [1, inf)

The final answer is the union of the two above answers.

Sum of an arithmetic series:  Sum = n( t₁ + t_n ) / 2

where          n = number of terms          t₁ = first term  = 1          t_n = last term  =  101

Since we don't know the number of terms, we can use this formula:  t_n  =  t₁ + (n - 1)d

where         d  =  common difference  =  4

t_n  =  t₁ + (n - 1)d     --->     101  =  1 + (n - 1)4     --->     101  =  1 + 4n - 4     --->     101  =  4n - 3

                                                        --->     104  =  4n     --->   n  =  26

Sum = n( t₁ + t_n ) / 2     --->     Sum = 26( 1 + 101 ) / 2     --->   Sum  =  1326

A problem with these questions:  In the questions they refer to the point R(4,-3), but on the graph they have plotted the point (3,-4).

In the questions, replace "R(4,-3)" with "R(3,-4)" and then try to solve them.

Question 1:  Take out a sheet of graph paper; plot the point R(3,-4). 

                     Draw the line through that point and the origin.

                     Draw the perpendicular to that line through the origin.

                     The resultant point will be on this second line, in the third quadrant.

                     The distance from the original point, R(3,-4), and the origin will be the same distance from the origin

                     to the point R' (and, I have no idea why they call it "V'").

Question 2: To reflect the point over the x-axis means that the new point will have the same x-value as the 

                    original point but it will be as far above the x-axis as the original point is below the x-axis.

Question 3: Start from the original point and count up 13 spaces. Where are you?

Question 4: This asks where the 4 endpoints of the kite end -- what is the only answer that has 4 points?

The sum of n integers is:  n(n + 1)/2.

Placing that into the problem:  [ n(n + 1)/2 ] / [ 2n ]  =  10

Multiplying both sides by 2n:     n(n + 1)/2  =  20n

Multiplying both sides by 2:          n(n + 1)  =  40 n

Simplifying:                                     n² + n  =  40 n

                                                    n² - 39n  =  0

Factoring:                                    n(n - 39)  =  0

Possible answers:                     n = 0     n = 39

Obviously, 0 isn't correct, so:  n = 39

One way (and I don't claim that this is the easiest way):

Draw a line segment from A perpendicular to CD ending at point X on CD.

Draw a line segment from B perpendicular to CD ending at point Y on CD.

You now have triange(ADX), rectangle(AXYB), and triangle(BYC).

Combine triangle(ADX) and triangle(BYC) so that points A and B coincide and points X and Y coincide, creating a triangle

with sides of lengths 13, 15, and 14.  [CY + DX = CD - 10 = 14]

Using Heron's formula to find the area of this new triangle:  s = (13 + 14 + 15)/2 = 21

--->     Area  =  sqrt[ 21(21-13)(21-14)(21-15) ]  =  84.

To find the height of this triangle (which is the width of the rectangle):

--->     Area  =  ½·base·height     --->     84  =  ½·(14)·height   --->     height  =  12.

Therefore, the area of the rectangle is:  10·12  =  120

and the total area is 84 + 120  =  204