While flying a kite at the beach, you notice that you are 100 yards from the kite’s shadow, which is directly beneath the kite. you also know that you have let out 150 yards of string. how high is the kite?

Alright so here you would use the Pythagorean theorem. a^2+b^2=c^2. 
You can create a straight line from the kites shadow to where you are standing. Then you can also create a straight line from the string you are holding to where it is attached to the kite.
lastly, you can draw a last line from the kite in the sky to its shadow on the ground. 
you have created a triangle. 
you know that from the string in your hand to where it is attached to the kite is 150 yards, this is also the hypotenuse. 
you also know that you are 100 yards from the kites shadow on the ground. 
this is one of the legs. 
so your equation would look something like 100^2+b^2=150^2

simplify to 10000+b^2=22500
subtract 10000 from each side
b^2=12500
square root both sides
your answer is approximately 111.8 yards 
 

Using the rise/run rule. you would get 2/6.simp. 1/3