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Yellow Δ area = √3        =>  3√3

Red   Δ area   = =>       => 25√3

White trapezoid area    => 33√3

__________________________

Total area = =>             => 61√3 square units

*****

Thanks

It's solved now. :)))

=^._.^=

-7x = 4x + 22

-11x = 22

x = -2

y = 14

(-2, 14)

I hope this helped. :))

=^._.^=

There are 25 possible values of n.

Using CRT + MMI

i=0;j=0;m=0;t=0;a=( 7,8, 9, 5);r= ( 0, 1, 2, 9);c=lcm(a); d=c / a[i];n=d % a[i] ;loop1:m++; if(n*m % a[i] ==1, goto loop, goto loop1);loop:s=(c/a[i]*r[j]*m);i++;j++;t=t+s;m=0;if(i< count a, goto4,m=m);printc,"m + ",t % c;return

N =2520m  +  2009, where m =0, 1, 2, 3........etc.

When m=0, then the smallest N that satisfies all 6 congruences ==2009

The first thing to notice is that the triangles are similar, and right triangles. 

BC = 2sqrt(6)

ABC side ratios are 

5 : 2sqrt(6) : 7

DAC side ratios are

? : 7 : x, where x = CD

Looking for x, 7/(2sqrt(6))*7 = 49sqrt(6)/12

I hope this helped. :)))

=^._.^=

Thank you so much, have a good life

Thank you, I have a question. Can you simplify this?

Hi Guest!

To solve this question, you can use case work. :))

4 digit plus 1 digit

abcd + e = 2247

*note abcd is written as 1000a + 100b + 10c + d. 

dcba + e = 4353

4 digit plus 2 digit

abcd + ef = 2247

dcba + fe = 4353

4 digit plus 3 digit

abcd + efg = 2247

dcba + gfe = 4353

4 digit plus 4 digit

abcd + efgh = 2247

dcba + hgfe = 4353

Try solving these cases by yourself. I wish you best of luck. :)))

If you need help, feel free to respond and I'll try to help more. 

=^._.^=

Kitsoundwave  could have carried a little  farther

45  / 24   =    (15*3)  / (8*3)  =  15  / 8

But, no big deal....

6^x   = 5     ⇒ (6^x)^3  = 5^3    ⇒  6^(3x)   = 125

6^ (3x + 2)  =   6^(3x) *6^2   =   36 * 6^(3x)   =  36 * 125   =  4500

(3.1  /  6.3)   ( 10^17  /10^11)   =

0.4920634920634921  *   10^6   =

4.92  x 10^5      times

Sorry....typo

    (0,1) U (4,+inf)  

x^(1/3)  + y^(1/3)   =  17       square both sides

(x^1/3  + y^1/3)^2   = 17^2

(x^2/3  + 2(xy)^1/3  + y^2/3)  =  289

x^2/3  + y^2/3  =  289   - 2(xy)^ (1/3)

Factor x + y as a sum of  cubes

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

3383  =(17)  ( x^(2/3)  - (xy)^(1/3)  + y^(2/3)  )

3383  /17    = x^(2/3) + y^(2/3)  - (xy)^(1/3) 

199  =  289  - 2(xy)^(1/3 )  - (xy)^(1/3)

189 - 289   =  - 3(xy)^(1/3)

-90  = -3(xy)^(1/3)

-90  / -3  =  (xy)^1/3

30  =  (xy)^(1/3      cube both sides

27000   =xy

By similar triangles, x = 11/2.

I originally went with n = 2 but I had overlooked the "but not all integer values" stipulation.  So never mind. 

incorrect :(

x ( x^2 - 5x+4) > 0

x(x-4)(x-1) >0                        (0,1) U (1,+inf)     (by graphing ....easiest)

Thanks!

bytes/pixel   * pixel = bytes     (see how 'pixel' cancels out?)

4 x 10⁰    *   2.2 x 10⁶ =   8.8 x 10⁶   bytes

Angle 14  =Angle  8   

Angle  7 =  180    - Angle 8  =  180 -  119     =  61

Angle 6 =  180   - Angle  3 =    180 - 140  = 40 

Angle  9  =   180  - Angle 7  - Angle  6  =   180 -  61  - 40  =    79   = Angle 11

Angle 16 =  180 - Angle 14  =   180  - 114   =  66

Angle 20  =   180 - Angle 11  - Angle  16  =   180  -  79  - 66  =  35  =  Angle  18

UV  is  the midline

Its  length = (sum of base lengths)   / 2 =   ( 44 + 78)   /2  =  ( 122)  / 2   =  61

We  have  the following relationship  set up by the bisected angle LMJ

JM  /LM   = JK/ KL

60  /75  = JK / KL

4/5  = JK / KL

JK   =  (4)  /(4 + 5) LJ  =    (4/9) (45)  =  20