Question 674652
What is the number of square units in the area of a triangle whose sides are 5, 6 and &#8730;<span style="text-decoration: overline">61</span>.  Explain answers in simplest form.  (I think it's 21, but i think it's wrong.)


<pre>
This is a right triangle because it satisfies the Pythagorean theorem

a = 5, b = 6 and c = &#8730;<span style="text-decoration: overline">61</span>

a² + b² = c²
5² + 6² = (&#8730;<span style="text-decoration: overline">61</span>)²
25 + 36 = 61
     61 = 61

Therefore it looks like this:

{{{drawing(200,200,-1,7,-1,7,triangle(0,0,5,0,5,6),
rectangle(4.7,0,5,.3),
locate(2.5,0,5), locate(5.1,3.3,6), locate(1.4,3.4,sqrt(61)) 
 )}}}

Therefore its base b = 5 and its height h = 6.  The formula
for the area of a triangle is  

             A = {{{1/2}}}bh
             A = {{{1/2}}}(5)(6)
             A = {{{1/2}}}(30)
             A = 15 square units.

The way to check it is to draw a 5×6 rectangle:

{{{drawing(200,200,-1,7,-1,7,rectangle(0,0,5,6),

locate(2.5,0,5), locate(5.1,3.3,6)  
 )}}}

Realize that the triangle above is one-half of its area 
because if you draw the diagonal it cuts the 5×6 rectangle 
into two right triangles just exactly like the one above.

{{{drawing(200,200,-1,7,-1,7,rectangle(0,0,5,6),
triangle(0,0,5,0,5,6),
locate(2.5,0,5), locate(5.1,3.3,6)  
 )}}}

and if you tile it off into square units like this:

{{{drawing(200,200,-1,7,-1,7,rectangle(0,0,5,6),
triangle(0,0,5,0,5,6),
locate(2.5,0,5), locate(5.1,3.3,6),rectangle(0,1,5,2),rectangle(0,2,5,3),
rectangle(0,3,5,4),rectangle(0,4,5,5), rectangle(1,0,2,6),rectangle(2,0,3,6),
rectangle(3,0,4,6)


 )}}}

You can see that there are 30 square units and since the
triangle has half as many square units of area as the rectangle,
then the triangle must have half of 30 square units of area or 15
square units of area.

Edwin</pre>