If the area of triangle ABC is 27, what is the value of p?

Maybe someone has a better way (easier) to do this, ACG...but here goes nothing....[note....it looks difficult, but as we shall see, once simplified, the math is fairly straightforward....!!!  ]

Let CB  be the base of the triangle....its length is √ [12*2 + p^2 ]

Slope  of CB  = -p/12

Equation  of  line  containing CB  is

y =( -p/12) x + p    multiply  through by 12

12y=  -px + 12p       we want the form  Ax  + BY + C  = 0...so....

px + 12y - 12p  = 0

The distance from this line to (2,12)  is the altitude of the triangle and is given by

l  p(2)  + 12(12) - 12p l  / √[ p^2 + 12^2 ]

So...the area is given by

27  = (1/2) CB  * altitude      ....and we  have.....

27  = (1/2) √ [12*2 + p^2 ] * l  p(2)  + 12(12) - 12p l  / √[ p^2 + 12^2 ]   ....simplify....

54  =  l  p(2)  + 12(12) - 12p l

54  =  l -10p + 144 l     

We have these two equations

-54  = -10p + 144 ⇒  p  = 99/5....reject because  p must be  < 12

Or...this equation....

54 = -10p + 144      subtract  144 from both sides

-90  = -10p

9  = p

Here's the  proof that this is correct : https://www.wolframalpha.com/input/?i=area+of+triangle++%5B+(0,9),+(2,12),+(12,0)+%5D

A second method...   

Slope of far right line is -12/10   it will intersect the y axis at 14.4

then the area formed by this largest triangle is

1/2  ( 14.4 x 12)      then subtract the little we-just-added triangle

- 1/2  (2.4 x 2)                            and the little triangle in the diagram

-1/2   (2 x (12-p))                          and the medium sized triangle in the lower left

-1/2  (p x 12)                                       and what is left = 27 (given)

86.4 -    2.4 -    12 + p    - 6p   = 27

72 - 5p = 27

-5p = -45

p = 9