CPhill

Draw a unit circle at the origin, O, and let P and Q loe pn this  circle

tan OP  = 4/1

cos OP = 1/sqrt(4^2 + 1^2)  = 1/sqrt 17      sin OP  = 4/sqrt 17

tan  OQ  = 5/1

cos OQ = 1/sqrt (5^2 + 1^2)  = 1/sqrt 26     sin QP = 5/sqrt  26

We can write P  as ( 1/sqrt 17 , 4/sqrt 17)

We can write Q as ( 1/sqrt 26 , 5/sqrt 26)

Slope  PQ  =

(5/sqrt 26 - 4/sqrt 17)  / ( 1/sqrt 26 - 1/sqrt 17)  =

(5sqrt 17- 4sqrt 26) / ( sqrt 17 - sqrt 26)  =

( 5 sqrt17 -4sqrt 26) (sqrt 17 + sqrt 26) / (17 -26)  =

(5*17 + sqrt 26 * sqrt 17 - 4*26) / -9

(-19 + sqrt 442) / -9 = 

(sqrt 442 - 19) / 9   ≈  -0.224

AB = 4  ???

https://web2.0calc.com/questions/geometry_50437

69 / 2553 = x / 1110

1110 * 69  = 2553 * x

x = 1110 * 69 / 2553   =  30

https://web2.0calc.com/questions/geometry_38859

Shortest altitude is drawn to the longest side

Area  =  sqrt [ 17 * (17 - 15) *(17 -10) * (17 - 9) ] = sqrt [17 * 2 * 7 * 8] = 4sqrt (119)

4sqrt (119)  = (1/2) 15 * shortest altitude

(8/15)sqrt (119) = shortest altitude ≈  5.82

See a similar problem here : 

https://web2.0calc.com/questions/trapezoid_85

log_a (ab)^2 =

2 log_a (ab) =

2 log_a(a) + 2log_a (b)  =

2 (1) + 2log_a (b) =

2 ( 1 +  log b / log a)

See a similar problem here :  https://web2.0calc.com/questions/geometry_136

https://web2.0calc.com/questions/geometry-question_42617

a + b = 4          square both sides

a^2 + 2ab + b^2 = 16

(a^2 + b^2) + 2ab =16

6 + ab + 2ab = 16

3ab = 10

ab = 10/3

a^2 + b^2 = 6 + ab

a^2 + ^2 - ab = 6

a^3 + b^3  = ( a + b) ( a^2 + b^2 - ab) =  (4)(6)  = 24

See a very similar problem  here with Heureka's excellent  approach

https://web2.0calc.com/questions/geometry_136

Call the first term a1   and the common difference, d

We gave these two equations

a1 + 22d = 1/4

a1 + 32d = 1/5      subtract the second equation from the first

-10d = 1/20

d = -1/200

To find the 43rd term :

a43 = a33 + 10d

a43 = a33 + 10 (-1/200)

a43 = 1/5 - 10/200

a43 = 1/5 - 1/20

a43 = 3/20

Only the odd terms can be positive.....so (-8/9)^(n-1)  = (8/9)^(n-1)  when n is odd

We need to solve this

1000 (8/9)^(n -1) > 1        divide both sides by 1000

(8/9)^(n-1) > 1/1000          take the log of both sides

log (8/9)^(n-1) > log (1/1000)    and we  can write

(n-1) log (8/9) > log (1/1000)   rearrange as

n <  log ( 1/1000) /log (8/9) + 1     we divided by a negative - log (8/9) - so, reverse the inequality sign

n <  ≈ 59.6

The 59th term is the last one > 1

So....the number of terms > 1  =   60/2 = 30

Let P = (x,y)

PA^2 + PB^2 + PC^2  =

(x -2)^2 + ( y- 4)^2  + (x + 3)^2 + (y -1)^2 + (x-1)^2 + ( y-7)^2 ....simplifying, we  have

3x^2 +3y^2 -24y + 80   complete the square on y

3[ (x-0)^2 + y^2  - 24y  + 144] + 80 - 3(144)  =

3 [ ( x-0)^2 + (y- 12)^2 ] - 352  =  3PQ^2 + k

Q= (0 , 12)

k = -352

A and B are endpoints of the diameter

The radius of the circle =1

So AB  = 2

Note that [BEFA] = [CEFD]

The line is y= 2x 

Any line of the form y = ax will split the rectangle ( a square) into = areas and will have a y intercept  = 0