CPhill

We can use  the  tangent-secant theorem here

This says that  the square  of the tangent segment  =   the length of the external part of the secant segment * the  length of the secant 

We  have  that

8^2  =  4 ( 4 +x)  simplify

64  = 16 + 4x      subtract  16 from both sides

48 = 4x     divide both sides by 4

12  =  x

cos (arctan (2))

Let  arctan (2)  = θ

This implies that

tan θ = 2  =  2  / 1    =   y  / x

And  cos θ  =  x / r

And r =   sqrt  ( x^2  + y^2)   sqrt  (1^2 + 2^2)  = sqrt (5)

So

cos (arctan (2))  =  

cos θ  =      1  /sqrt (5)  =   sqrt (5) / 5

Note  that arc ADC  =  2*measure of angle AEC =  2*110  = 220°

So  angle ADC  intercepts  arc AEC =  360 - 220  = 140°

So  we have this relationship : 

x = (1/2) (arc ADC - arc AEC)  =  (1/2) ( 220 - 140) =  (1/2)(80)  = 40° 

OK...no prob   !!!!

If we draw radii to the tangent points we will have a quadrilateral  with two interior angles  = 90°  and  another interior angle =  central angle = 120°

And   the interior angles of a quadrilateral  sum to 360°

So

90 + 90 + 120   +  x  =  360

300 + x  = 360

x =  360 - 300  =  60°

We can solve this

0  ≤ [ -2x + 5] / 7  <  1       multiply both sides by 7

0 ≤  -2x + 5  <   7

We have two inequalities

0 ≤ -2x + 5                            -2x + 5 < 7

               Subtract  5 from both sides

-5 ≤ -2x                                                              -2x < 2

                            Divide both sides by -2

                     Reverse the inequality sign         

5/2 ≥ x                                                                  x > - 1

So....the  solution interval for   this is    -1 < x  ≤ 5/2

We can use similar triangles to solve this....see the following image

Let AD  be  one side of the square

And let   DE  be the  other side of the square

And  using  similar riangles, we  have  that triangle  BDE is similar to triangle BAC

So

BD/ DE  =  BA/ AC

BD/ DE =  2 / 3

But DE = AD

So

BD / AD   =  2/3

BD : AD  = 2 : 3

So there are 5 equal parts of AB  and AD is 3 of them...so AD = (3/5)(AB)  = (3/5)(2) = 6/5  = side of the square

Nice, hugo   !!!!!!

Some data is missing.....

I think I remember this from high school

The Doppler effect  says   that the  pitch of the sound will  be  lower  as  the distance from the object increases

So  (A) is correct