Question 879410
{{{drawing(300,300,-5,7,-8,4, grid(1),

triangle(-4,-3,sqrt(27),-sqrt(48),4,3),
locate(4.3,3.4,"(4,3)"),
locate(-4.3,-3.3,"(-4,-3)"),locate(2,-7,

(matrix(1,3,sqrt(27),",",-sqrt(48))))


 )}}}
<pre>
We use the distance formula on all three sides to show
that all the sides have the same length.

Distance from (-4,-3), (&#8730;<span style="text-decoration: overline">27</span>, -&#8730;<span style="text-decoration: overline">48</span>)

{{{d}}}{{{""=""}}}{{{ sqrt( (x[2]-x[1])^2 + (y[2]-y[1])^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( ((sqrt(27))-(-4))^2 + ((-sqrt(48))-(-3))^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( (sqrt(27)+4))^2 + (-sqrt(48)+3)^2)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(27+8sqrt(27)+16 + 48-6sqrt(48)+9)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 + 8sqrt(9*3)-6sqrt(16*3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 + 8*3sqrt(3)-6*4sqrt(3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 + 24sqrt(3)-24sqrt(3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100)}}}

{{{d}}}{{{""=""}}}{{{10}}}

-------------------------------

Distance from (-4,-3) to (4,3)

{{{d}}}{{{""=""}}}{{{ sqrt( (x[2]-x[1])^2 + (y[2]-y[1])^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( ((4)-(-4))^2 + ((3)-(-3))^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( (4+4)^2 + (3+3)^2)}}}

{{{d}}}{{{""=""}}}{{{ sqrt( (8)^2 + (6)^2)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(64 + 36)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100)}}}

{{{d}}}{{{""=""}}}{{{10}}}

--------------------------------------

Distance from (4,3) to (&#8730;<span style="text-decoration: overline">27</span>, -&#8730;<span style="text-decoration: overline">48</span>)

{{{d}}}{{{""=""}}}{{{ sqrt( (x[2]-x[1])^2 + (y[2]-y[1])^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( ((sqrt(27))-(4))^2 + ((-sqrt(48))-(3))^2)}}} 

{{{d}}}{{{""=""}}}{{{ sqrt( (sqrt(27)-4))^2 + (sqrt(48)-3)^2)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(27-8sqrt(27)+16 + 48+6sqrt(48)+9)}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 - 8sqrt(9*3)+6sqrt(16*3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 - 8*3sqrt(3)+6*4sqrt(3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100 - 24sqrt(3)+24sqrt(3))}}}

{{{d}}}{{{""=""}}}{{{ sqrt(100)}}}

{{{d}}}{{{""=""}}}{{{10}}}

All three sides are equal to 10, so the triangle is equilateral.

Edwin</pre>