CPhill

Third one  

Note that arc JK + arc KL  =180

Therefore

(5x + 23)  + (17x - 41)  = 180      simplify

22x  - 18  = 180         add 18 to both sides

22x  = 198       divide both sides by 22

x  = 9

arc JK  =  5x + 23   =   5(9) + 23  =  45 + 23  =  78°

Since angle MVJ  = angle LVK,  arc MJ  =  arc KL  =  17x - 41  = 17(9) - 41  =  112° 

arc LMK  =  arc LMJ + arc JK   =  180 + 78  = 258°

Second one  [ at the top right ]

Notice that angle QUR  is a vertical angle to angle TUS.....so its measure is also  = 19°

And the measure of angle  PUR = 90°

So the measure of  arc PQ  =  measure of angle  PUQ  =  measure of  angle PUR - measure of angle QUR  = 

90  - 19   =   71°

Measure of arc SR   =   180  - 19   =  161°

Measure of arc QRT  =  measure of arcs SR + QR =  180 + 19   =  199° 

Measure of arc  PSR  =  measure of arc PT  + measure of arc TSR = 90  + 180  = 270°  

Measure of  arc PS   =   measure of arc PT  + measur of arc TS  = 90  + 19    =  109°

First one.....

Note that central angle CHD  is right  = 90°

And the measure of  arc  CD  has the same measure  = 90°

Angle DHE  also = 90°  [ since it is supplemental to angle CHD]

So arc  DE has the same measure  = 90°

And arc  FE = 34°

So...arc  FD  [I'm asssuming arc FED ]  is the sum of these two arcs  =  124° 

The measure of arc DCF  =  360  -  measure of arc FD   =  360 - 124  = 236°

The measure of arc GDF   = measure of arc GCD  +  measure of arc FED  =  180 + 124  = 304°

A.  81

B.  9

No prob....

25m^2  x   [ 1 yd^2] / [0.8 m^2 ]   =  [25 / 0.8 ] yd^2  =  31.25 yd^2

We can construct the following rectangle  ABCD  with squares of side 1

The length of the diagonal  of  one of these squares =  √[1^2 + 1^2] = √2

And the diagonal AC of the image  = √[6^2 + 8^2]  = √100  =  10

Since these are similar figures, we can solve for the diagonal length of your problem, AC, thusly

6 / √2 =    AC / 10         cross-multiply

10*6 / √2  =  AC    =    60/√2  =  60√2/ 2   =  30√2

It can't be "B"  because we need the entire  range possible  of  (a+b)^2

"B"  says  that   (a + b)^2   must be  >  25    [ but   < 36 ]

But this isn't  true......note  if   a = 0   and  b  = 0....then (a + b)^2  = (0 + 0)^2  =  0

So....we can make   (a +b)^2   less than 25

Let   a  =   m + ni

Let  b conjugate =     p  - qi  

So

a* (b conjugate)  =   (m + ni) (p - qi)  =  mp + (np - mq)i  + nq    =  (mp + nq) + (np - mq) i =   -1 + 5i 

mp + nq  =  -1

np - mq  = 5

And

a conjugate  =  m - ni

b =  p + qi

So

(a conjugate) * b  =    ( m - ni) ( p + qi)  = mp + (mq - np)i + nq  = ( mp + nq)  + (mq - np)i 

So

(mp + nq)  =  -1

(mq - np)  =  -5

So

(a conjugate)  * b   =    -1  - 5i

Note that   (a + b)^2   either produces a positive result   or  0

So.....the least that (a + b)^2  could be is  0

And the greatest that it can be  is if  a ≈ 4  and b  ≈ 2

So    (a + b)^2  =  ( ≈4  + ≈2) ^2   =  (≈ 6)^2  =  ≈ 36

Therefore

0 ≤ (a+b)^2 < 36