Question 176130
<pre><font size = 4 color = "indigo"><b>

For this line PS to be the hypotenuse of an isosceles
right triangle,
 
{{{drawing(289,289,-10,2,-5,7,
 locate(-6,-2,"P(-6,-2)"),locate(-6,5.6,"S(-6,5)"),
graph(289,289,-10,2,-5,7),
 
line(-6,-2,-6,5) )}}}
 
We could have two possibilities for such an
isosceles right triangle. This is one 
possibility for T, call it little "t".
 
{{{drawing(289,289,-10,2,-5,7,
 
graph(289,289,-10,2,-5,7), locate(-6,-2,"P(-6,-2)"),
line(-2.5,1.5,-6,5), line(-2.5,1.5,-6,-2),locate(-6,5.6,"S(-6,5)"),
line(-6,-2,-6,5),locate(-2.5,1.5,t) )}}}
 
This is another possibility for T, we'll
call it big T.
 
{{{drawing(289,289,-10,2,-5,7,
 
graph(289,289,-10,2,-5,7), locate(-6,-2,"P(-6,-2)"),locate(-9.7,1.5,T),
line(-9.5,1.5,-6,5), line(-9.5,1.5,-6,-2),locate(-6,5.6,"S(-6,5)"),
line(-6,-2,-6,5) )}}}
 
Put them together side by side and you have
a square:
 
{{{drawing(289,289,-10,2,-5,7,
line(-9.5,1.5,-6,5), line(-9.5,1.5,-6,-2),
graph(289,289,-10,2,-5,7), locate(-6,-2,"P(-6,-2)"),locate(-2.5,1.5,t),
line(-2.5,1.5,-6,5), line(-2.5,1.5,-6,-2),locate(-6,5.6,"S(-6,5)"),locate(-9.7,1.5,T),
line(-6,-2,-6,5) )}}}
 
Now from P to S is 7 units. That means that the diagonal of 
the square is 7 units long.  We also know that both diagonals
have the same length, so let's draw in the other diagonal Tt,
from the T on the left to the t on the right, crossing the other
diagonal PS at their common midpoint X: 

{{{drawing(289,289,-10,2,-5,7,
line(-9.5,1.5,-6,5), line(-9.5,1.5,-6,-2),
graph(289,289,-10,2,-5,7), locate(-6,-2,"P(-6,-2)"),locate(-2.5,1.5,t),
line(-2.5,1.5,-6,5), line(-2.5,1.5,-6,-2),locate(-6,5.6,"S(-6,5)"),locate(-9.7,1.5,T), line(-9.5,1.5,-2.5,1.5), locate(-6,1.5,X),
line(-6,-2,-6,5) )}}} 

Tt has to also be 7 units long. because PS is.
That makes Xt be half of that or 3.5 units long.
Now X is 6 units horizontally away from the y-axis
and since Xt = 3.5 units, t has to be 6-3.5 or 2.5
units from the y-axis.  That means the x-coordinate 
of t has to be -2.5.  

We know what the y-coordinate of t is because it's 
the same as the y-coordinate of X.  That's the 
midpoint of P(-6,-2) and P(-6,5), or X(-6,1.5), and 
so t is the point t(-2.5,1.5).

Use the same reasoning and you'll get that the 
coordinates of T are T(-9.5,1.5)

Edwin</pre>