If these three segments are connected end-to-end to form a triangle, will the triangle be acute, right, or obtuse?

 


Correct answer: Obtuse

Let's start by checking if it's a right triangle. If it were right, then the longest side (length 8) would need to be the hypotenuse, \LARGE C and the other two sides (lengths 5 and 6) would need to be the two legs, \LARGE a   and \LARGE b which meet at a right angle.

So we can test if the triangle is right by checking if the Pythagorean identity is true given the lengths of the three sticks: does \LARGE a^2+b^2=c^2?

We have \LARGE a^2+b^2+5^2+6^2=25+36=61  

Therefore, the triangle is not right.

Also, we can see from these calculations that  \LARGE c^2>a^2+b^2.

This means that the triangle will be obtuse because the "extra" length of the longest side stretches the opposite angle to a measure greater than \LARGE 90^0 

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