Question 1196827
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A right triangle has a hypotenuse of 1. One leg is x. The other leg is (x - 7).
Find the area of the triangle. 
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Such a triangle does not exist.



Proof:    the difference of two sides,  x  and  x-7,  must be less than the third side  (from the triangle's inequality).


            But the difference of the sides  x  and  (x-7)  is  7,  while the third side is  1.


            Contradiction.



As it is worded,  printed and posted,  this   "problem "  is dead just from the very moment of its creation.



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<U>Comment from student</U> : You are right. It was a typo. Here it is: 
Hypotenuse = x + 1 One side = x Other side = x - 7 Find the area.



<U>My response</U> :  It is just another story.  See my solution below.


<pre>
Write the Pythagorean equation

    x^2 + (x-7)^2 = (x+1)^2.


Simplify and find x

    x^2 + (x^2 - 14x + 49) = x^2 + 2x + 1

    x^2 - 16x + 48 = 0

    (x-4)*(x-12) = 0.


For x, we have two roots:  x= 4  and  x= 12.


The root x= 4 is not the solution, since then  x-7  is negative.


The root x= 12 produces the right angled triangle (5,12,13).


Its area is  {{{(1/2)*5*12}}} = 5*6 = 30 square units.    <U>ANSWER</U>
</pre>

Solved.