The base of a triangular sign exceeds the height by 8 inches. if the area of the sign is 24 square inches, find the length of the base and the height of the triangle.

The height is:  "4 inches" ;
 and the base length is:  "12 inches" .

Explanation:

The formula of the area of a triangle is:

Area = (1/2) * (base length) * (perpendicular height) ; or write as:

A = (1/2) * (b) * (h) ;

 Given:  A = 24 in² ;
              b = h + 8 ;

 Find:  "b" ;  and:  "h" ;

 →  Since:  " A = (1/2) * b * h " ; 

     →  Plug in our known values:
     
            →  24 = (1/2) * (h + 8) * h ;

Find:  h ;
Find  "b" , which is: "(h +8)" ;

We have:  

  →  24 = (1/2) * (h + 8) * h ;

Multiply EACH SIDE of the equation by "2" ; to get rid of the fraction:

  → 2 * {24 = (1/2) * (h + 8) * h} ;
 
     → to get:

     → 48 = 1 * (h + 8) * h ;

↔ Rewrite:  h(h + 8) = 48 ;

Note the "distributive property of multiplication":

    a(b+c)  = ab + ac ;
  
    a(b−c)  = ab − ac ;

  →  So;  h(h + 8) = h*h  + h*8 = h² +8h = 48 ;

We have:  " h² + 8h = 48 " ;  To solve for "h" ;  let us see if we can
          write this equation in "quadratic format" ; that is:

    " ax² + bx + c = 0 ;  a ≠ 0 ; " ;

We have:   h² + 8h = 48  ;  Subtract "48" from EACH SIDE of the equation:

                  h² + 8h − 48 = 48 − 48 ;

 to get:    h² + 8h − 48 = 0 ;

Note that is equation IS, in fact, written in "quadratic format" ;
   that is: "ax² + bx + c = 0 ;  a ≠ 0 " ;  
 
      in which:  a = 1 ;
(Note:  The "implied coefficient" of "1"; since anything multipled by "1" is that same result);
                      b = 8 ;
                      c = - 48;

Now, let us see if we can solve by factoring; if we cannot, we can use the quadratic equation formula:

Let us trying factoring:  h² + 8h − 48 = (h+12) (h − 4) = 0 ; 

Since anything multiplied by "zero" equals "zero" ; 

Then either:  (h+12) = 0 ;  h = -12 ; 
                     (h − 4) = 0 ;  h = 4 ;
 
So we have two (2) values for "h" ;  "h = 4" , and "h = -12" .

So, which value do we use for "h"?  Since "h" refer to "height"; 
we know that "height" cannot be a "negative value";  so we use: 

"h = 4" .

Now, we are given:  "b = h + 8 = 4 + 8 = 12 "

So,  h = 4 ; b = 12.

Now check our work:  "A = (1/2) (b) (h)" ; Given "A = 24" .

24 = (1/2) (12) (4)?  24 = (1/2) * 48 ?  YES!

So, the height is:  "4 inches" ;
 and the base length is:  "12 inches" .
 

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