SOLUTION: PS is the hypotenuse of isosceles right triangle PST for P(-6,-2) and S(-6,5). Find some possible coordinates of T.

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Question 176130: PS is the hypotenuse of isosceles right triangle PST for P(-6,-2) and S(-6,5). Find some possible coordinates of T.
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!


For this line PS to be the hypotenuse of an isosceles
right triangle,
 
drawing%28289%2C289%2C-10%2C2%2C-5%2C7%2C%0D%0A+locate%28-6%2C-2%2C%27P%28-6%2C-2%29%27%29%2Clocate%28-6%2C5.6%2C%27S%28-6%2C5%29%27%29%2C%0D%0Agraph%28289%2C289%2C-10%2C2%2C-5%2C7%29%2C%0D%0A+%0D%0Aline%28-6%2C-2%2C-6%2C5%29+%29
 
We could have two possibilities for such an
isosceles right triangle. This is one 
possibility for T, call it little "t".
 
drawing%28289%2C289%2C-10%2C2%2C-5%2C7%2C%0D%0A+%0D%0Agraph%28289%2C289%2C-10%2C2%2C-5%2C7%29%2C+locate%28-6%2C-2%2C%27P%28-6%2C-2%29%27%29%2C%0D%0Aline%28-2.5%2C1.5%2C-6%2C5%29%2C+line%28-2.5%2C1.5%2C-6%2C-2%29%2Clocate%28-6%2C5.6%2C%27S%28-6%2C5%29%27%29%2C%0D%0Aline%28-6%2C-2%2C-6%2C5%29%2Clocate%28-2.5%2C1.5%2Ct%29+%29
 
This is another possibility for T, we'll
call it big T.
 
drawing%28289%2C289%2C-10%2C2%2C-5%2C7%2C%0D%0A+%0D%0Agraph%28289%2C289%2C-10%2C2%2C-5%2C7%29%2C+locate%28-6%2C-2%2C%27P%28-6%2C-2%29%27%29%2Clocate%28-9.7%2C1.5%2CT%29%2C%0D%0Aline%28-9.5%2C1.5%2C-6%2C5%29%2C+line%28-9.5%2C1.5%2C-6%2C-2%29%2Clocate%28-6%2C5.6%2C%27S%28-6%2C5%29%27%29%2C%0D%0Aline%28-6%2C-2%2C-6%2C5%29+%29
 
Put them together side by side and you have
a square:
 
drawing%28289%2C289%2C-10%2C2%2C-5%2C7%2C%0D%0Aline%28-9.5%2C1.5%2C-6%2C5%29%2C+line%28-9.5%2C1.5%2C-6%2C-2%29%2C%0D%0Agraph%28289%2C289%2C-10%2C2%2C-5%2C7%29%2C+locate%28-6%2C-2%2C%27P%28-6%2C-2%29%27%29%2Clocate%28-2.5%2C1.5%2Ct%29%2C%0D%0Aline%28-2.5%2C1.5%2C-6%2C5%29%2C+line%28-2.5%2C1.5%2C-6%2C-2%29%2Clocate%28-6%2C5.6%2C%27S%28-6%2C5%29%27%29%2Clocate%28-9.7%2C1.5%2CT%29%2C%0D%0Aline%28-6%2C-2%2C-6%2C5%29+%29
 
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%28289%2C289%2C-10%2C2%2C-5%2C7%2C%0D%0Aline%28-9.5%2C1.5%2C-6%2C5%29%2C+line%28-9.5%2C1.5%2C-6%2C-2%29%2C%0D%0Agraph%28289%2C289%2C-10%2C2%2C-5%2C7%29%2C+locate%28-6%2C-2%2C%27P%28-6%2C-2%29%27%29%2Clocate%28-2.5%2C1.5%2Ct%29%2C%0D%0Aline%28-2.5%2C1.5%2C-6%2C5%29%2C+line%28-2.5%2C1.5%2C-6%2C-2%29%2Clocate%28-6%2C5.6%2C%27S%28-6%2C5%29%27%29%2Clocate%28-9.7%2C1.5%2CT%29%2C+line%28-9.5%2C1.5%2C-2.5%2C1.5%29%2C+locate%28-6%2C1.5%2CX%29%2C%0D%0Aline%28-6%2C-2%2C-6%2C5%29+%29 

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