Question

suppose someone tells you that she has a triangle with sides having lengths 2.6,8.1 and 8.6. Is this a right triangle? why or why not? Is there an angle in the triangle larger than 90 degree?

Answer 1

If you're only provided with the lengths of a triangle, and you're asked to determine whether or not the triangle is right or not, you'll need to rely on the Pythagorean Theorem to help you out. In case you're rusty on it, the Pythagorean Theorem defines a relationship between the legs of a right triangle and its hypotenuse, the side opposite its right angle. That relationship is a² + b² = c², where a and b are the legs of the triangle, and c is its hypotenuse. To see if our triangle fits that requirement, we'll have to substitute its lengths into the equation.

How do we determine which length is the hypotenuse, though? Knowledge that the hypotenuse is always the longest length of a right triangle helps here, as we can clearly observe that 8.6 is the longest we've been given for this problem. The order we pick the legs in doesn't matter, since addition is commutative, and we'll get the same result regardless of the order we're adding a and b.

So, substituting our values in, we have:

(2.6)² + (8.1)² = (8.6)²

Performing the necessary calculations, we have:

6.76 + 65.61 = 73.96
72.37 ≠ 73.96

Failing this, we know that our triangle cannot be right, but we do know that 72.37 < 73.96, which tells us something about what kind of triangle it is. Imagine taking a regular right triangle and stretching its hypotenuse, keeping the legs a and b the same length. This has the fact of increasing the angle between a and b. Since the angle was already 90°, and it's only increased since then, we know that the triangle has to be obtuse, which is to say: yes, there's an angle in it larger than 90°.