Question

The sides of a triangle are 77, 56, and 59. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.

Answer 1

Acute

Explanation:

The Pythagorean Theorem refers to
`a^(2) + b^(2) =c^(2)`, no (and only on right angled triangles)? We would have to use the cosine and sine rules to answer this question.

First, lets use the cosine rule to get an angle, then use the sine rule to get another, which would allow us to then determine the third, which will allow us to determine the type of triangle.

`cosA= (b^2+c^2-a^2)/(2bc)` → 
`cosA= (77^2+56^2-59^2)/(2(77)(56))` →
`cosA= (5584)/(8624)` →
`cos^(-1)(5584)/(8624) = A` →
`A = 49.65 = 50`~

Now we can use the sine rule to find the other angles:

`(sin50)/(59) = (sinB)/(77)` →
`77(sin 50)/(59) = sin B` →
`sinB = 0.9997529...` →
`sin^(-1) 0.9997529... = B = 88.726... = 89`~

So, if A = 50° and B = 89°, that would mean that the remaining angle is
`180 - 50-89=180-139=41`

A = 50°

B = 89°

C = 41°

As none of the angles exceed or are equal to 90°, it must be an acute triangle!