geno3141

I'm assuming that the problem is with the peak.

There are four congruent triangles, each with a base of 8.

We need to find the slant height of this pyramid; this will be the height of each triangle.

To find this length, consider a right triangle from the top point to the middle of the base and over to the middle of each side.

The length from the top point straight down to the middle of the base is 8.

The length from the middle of the base to the middle of one side is 4.

Therefore the slant height is the hypotenuse of this triangle:  c²  =  4² + 8²     --->     c  =  sqrt(80)

So, each triangle has an area of  ½·8·sqrt(80) and there are four of these triangles.

The base is 8 x 8.

Each side is 8 x 13.

Now, its calculator time ...

Seven positive integers, mean = 5, median = 4, and two modes 2 and 8.

Since there are an odd number of numbers and 4 is a median, at least one of the numbers is 4 and three of the numbers must be larger than 4 and three of the numbers must be smaller than 4.

Since both 2 and 8 are modes, either both are used twice or both are used 3 times.

When I tried to use them three times, I couldn't get anything to work.

So, using them twice, I have 2, 2, 4, 8, 8, giving a sum of 24.

For seven numbers to have a mean of 5, their sum must be 7 x 5  =  35.

I need another 11, split between a number smaller than 4 and a number larger than 4.

If I try 3 for the smaller number, this means that I have to use 8 for the larger number, but I can't because that would mess up the modes.

If the number is 1, than the other number must be 10.

1, 2, 2, 4, 8, 8, 10

The "power of the point":     DB · DC  =  DF · DE

                                 4 · [ 4 + (x - 3) ]  =  5 · [ 5 + (x - 6) ]

                                        4 · [ x + 1 ]  =  5 · [ x - 1 ]

                                                4x + 4  =  5x - 5

                                                        x  =  9

From this, you can find all the lengths ...

Pythagorean Theorem time!

For the triangle on the left:  a² + b²  =  c²     --->     1² + b²  =  5²     --->     b  =  sqrt(24)

For the griangle on the right:  5² + (sqrt(24) ² =  c²     --->     25 + 24 =  c²     --->     c  =  sqrt(49)    (which you can simplify)

To find a revolution of 270^o, I would do this:  the y-value of the initial point becomes the x-value after rotation

                                                                       the negative of the initial x-value becomes teh y-value after rotation.

A(3, -2)   --->   A'(-2, -3)

B(8, 2)   --->   B'(2, -8)

C(5, 10)   --->   C'(10, -5)

But, graph these to check to see if this works!

I'm assuming that the notation "T_1,-6" means a translation of 1 of the x-value and -6 of the y-value:

A(3, -2)   --->   A''(4, -8)

B(8, 2)   --->   B''(9, -4)

C(5, 10)   --->   C''(6, 4)

To see whether they are congruent or not, graph them and measure. My initial response is that they will be congruent because rotations and translation preserve distance and angles.

For the box on the left:

-- front side of 4 x 7

-- back side of 4 x 7

-- top side of 3 x 7

-- bottom side of 3 x 7

-- left side of 3 x 4

For the box on the right:

-- front side of 3 x 9

-- back side of 3 x 9

-- top side of 3 x 3

-- bottom side of 3 x 3

-- right side of 3 x 9

-- left side of 3 x 5

Find all the individual area and add them together for the final answer.

The conjugate of  [ sqrt(3) -1 ]  is  [ sqrt(3) + 1 ]

Numerator:  [ 6 + 10sqrt(3) ] · [ sqrt(3) + 1 ]  =  6sqrt(3) + 6 + 10sqrt(3)sqrt(3) + 10sqrt(3)

                                                                      =  6sqrt(3) + 6 + 10·3 + 10sqrt(3)

                                                                      =  16sqrt(3) + 36

Denominator:  [ sqrt(3) - 1 ] · [ sqrt(3) + 1 ]  =  3 - 1  =  2

Answer  =  [ 16sqrt(3) + 36 ] / 2  =  8sqrt(3) + 18

The cube root is the same as the 1/3 power.

(10⁴x)^(1/3)  =  10^4·(1/3) · x^(1/3)  =  10^4/3x^(1/3)  

[ 2x² + 2x - 2 ] / [ x(x² - 1) ]  =  [ 2x² + 2x - 2 ] / [ x(x + 1)(x - 1) ] 

To get each term to have the common denominator of x(x + 1)(x - 1),

--  multiply the first term:  A/x  by  [ (x + 1)(x - 1) ] / [ (x + 1)(x - 1)]

        so that the numerator becomes:  A(x + 1)(x - 1)  =  Ax² - A

--  multiply the second term:  B/(x - 1)  by  [ x(x + 1) ] / [ x(x + 1) ]

        so that the numerator becomes:  B(x)(x + 1)  =  Bx² + Bx

--  multiply the third term:  C/(x + 1)  by  [ x(x - 1) ] / [ x(x - 1) ]

        so that the numerator becomes:  C(x)(x - 1)  =  Cx² - Cx

All the denominator are  (x)(x + 1)(x - 1), so all we have to do is look at the numerators:

     2x² + 2x - 2  =   Ax² - A  +  Bx² + Bx  +  Cx² - Cx

     2x² + 2x - 2  =   Ax² +  Bx² +  Cx²  + BX - Cx - A 

Therefore:  Ax² +  Bx² +  Cx²  =  2x²      --->     (A + B + C)x²  =  2x²     --->     A + B + C  =  2

                                   Bx - Cx  =  2x       --->               (B - C)x  =  2x      --->            B - C  =  2

                                            -A  =  -2                                                        --->                  A  =  2

Solving, we get:  A = 2     B = 1     C = -1

1)  tan(theta)  =  y/x     --->     theta  =  tan^-1( -5sqrt(3) / 5)  =  2pi/3

      r  =  sqrt( (5)² + (-5sqrt(3))² )  =  10