<h2><u>Answers:</u></h2>
These problems can be solved by the Thales’s Theorem, which states:
<em>Two triangles are similar when they have equal angles and proportional sides  </em>
In addition, i<u>f two triangles are similar, their angles are similar as well</u>. This means that the relation between two sides of the big triangle is equal to the relation between two sides of the small triangle, as follows:  
 
 
Knowing this, lets’s begin with the answers:
1. For this first problem, see the <u>first figure attached</u>. There are shown the two triangles, and we are told both are similar, this means (according the prior explanation above) that we can use the Thale’s Theorem to find  .
.  
Therefore, we can establish a relation between two sides of the big triangle which is equal to the relation between two sides of the small triangle; in this case let’s use  and
 and  for the first triangle and
 for the first triangle and  and
 and  for the second:
 for the second:

 
    
Now we find  :
:
 
    
 
    
 
    
 
    
 
    
Finally, the value of  is 12 units.
 is 12 units.
2. For this problem, see <u>the second figure attached</u>. In order to find the values of the segments BE and EC, we will use two sides of each triangle (ABE and DCE) according to the Thale’s Theorem:
<u>For the segment BE:</u>


Simplifying:
 >>>>>This is the length of the segment BE
  >>>>>This is the length of the segment BE
<u>For the segment EC:</u>


Simplifying:
 >>>>>This is the length of the segment EC
>>>>>This is the length of the segment EC
3. For this problem, see the <u>third figure attached:</u>

Solving for  :
:

Finally:
 >>>>>This is the height of the smaller sail
>>>>>This is the height of the smaller sail
4. For this problem, see the <u>fourth figure attached.</u>

Solving for  :
:

 >>>>>This is the height from the floor to the son's hand
  >>>>>This is the height from the floor to the son's hand