Math CentralQuandaries & Queries


Question from audrey, a student:

The three sides of a right angles triangle measure x-2, x+5, and 2x-1 in length.
What are the possible lengths of the hypotenuse?

... I'm doing the equation c2=a2+b2 and subbing in the numbers but nothing makes sense

Hi Audrey,

I started by saying suppose the hypotenuse has length $2x - 1$ units. Then by Pythagoras Theorem

\[(x - 2)^2 + (x + 5)^2 = (2x - 1)^2\]

I then expanded this equation, simplified it and solved the resulting quadratic equation for $x.$ I found two solutions, one with $x$ positive and the other with $x$ negative. I then checked the positive value and found that the lengths did form a Pythagorean Triple which gave me a possible length for the hypotenuse. The negative value for $x$ gives $x - 2$ as negative and hence it can't be the length of the side of a triangle.

Now suppose the hypotenuse is of length $x + 5$ and perform the analogous procedure. Finally let the hypotenuse have length $x - 2$ and repeat the process.


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