geno3141

If the two numbers are  a  and  b:

arithmetic mean  =  (a + b) / 2      --->     (a +b) / 2  =  75     --->     a + b  =  150

geometric mean  =  sqrt(a·b)     --->     sqrt(a·b)   =  21     --->     ab  =  441     --->     b  =  441 / a

Combining:  a + 441 / a  =  150    --->     a² + 441  =  150a     --->     a² - 150a + 441  =  0

Using the Quadratic Formula:  a  =  [ 150 +/- sqrt( 20736) ] / 2     --->     147  or  3

a  is one of the two possibilities;  b  is the other.

The altitude to the hypotenuse will divide the original triangle into two isosceles right triangles.

Since the altitude is 4, the base of each of these two new triangles will also be 4, making the base of 

the original triangle 8.

Area  =  1/2 x 8 x 4  =  16

Let  z = a + bi

Then:

1)  |z - 1|  =  | (a + bi) - 1 |  =  | (a - 1) + bi |  =  sqrt( (a - 1)² + (b)² )

2)  |z + 3|  =  | (a + bi) + 3 |  =  | (a + 3) + bi |  =  sqrt( (a + 3)² + b² )

3)  |z - i|  =  | (a + bi) - i |  =  | a + (b - 1)i |  =  sqrt( (a)² + (b - 1)² }

Setting 1) and 2) equal to each other:

     sqrt( (a - 1)² + (b)² )  =  sqrt( (a + 3)² + b² )     --->    (a - 1)² + b²   =   (a + 3)² + b²  

        --->   a² - 2a + 1 + b²  =  a² + 6a + 9 + b²     --->     -8a  =  8     --->     a = -1

Setting 1) and 3) equal to each other:

     sqrt( (a - 1)² + (b)² )  =  sqrt( (a)² + (b - 1)² }     --->    (a - 1)² + (b)²  =  a² + (b - 1)²

       --->     a² - 2a + 1 + b²  =  a² + b² - 2b + 1     --->     a  =  b     --->     b = -1

Is there another answer?  

If BD both bisects AC and angle(B), then triangle(ABC) is an isosceles triangle with AB = BC.

Since AB = AC, AB = AC = BC, making the triangle equilateral.

Therefore, angle(A) = 60^o.

Since 44 inches of wire is 2·22 inches of wire, the cost will be 2·66 cents = 132 cents, or $1.32.

Let  A = 1,  B = 2, and  C = 3.

A₆ + B₆  =  C₆     --->     1₆ + 2₆  =  3₆

B₆ + B₆ + C₆  =  AA₆     --->     2₆ + 2₆ + 3₆  =  11₆  

ABC₆  =  123₆     which, in base ten, is:  1·6² + 2·6 + 3  =  51

Line 1:  5x + 8y  =  -9

                     8y  =  -5x - 9

                       y  =  (-5/8)x - 9/8

              The slope of this line is  -5/8.

               This means that the slope of any line perpendicular to it is 8/5.

Line 2:  point = (10, -12)     slope = 8/5

             Point-slope form for a line:  y - y₁  =  m(x - x₁)

                                                        y - -12  =  (8/5)(x - 10)

                                                        y + 12  =  (8/5)(x - 10)

                                                       5y + 60  =  8x - 80

                                                               5y  =  8x - 140

                                                                 y  =  (8/5)x - 28

Can you finish?

12y² - 65y + 42 - 25y  =  12y² - 90y + 42  =  (2y - 14)(6y - 3)

Can you finish? 

There are 16 possibilities, 4 for the numerator and 4 for the denominator.

But, some of these are duplicates; for example:  4/2  =  2/1

I would suggest that you write out the entire list and remove the duplicates.