Question

the sides of a triangle have lengths 6, 27, and 29. what kind of triangle is it?
acute
right
obtuse

Explanation:

Step1: Recall the triangle inequality theorem for types of triangles

For a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), we use the following:

• If \(a^{2}+b^{2}=c^{2}\), the triangle is right.
• If \(a^{2}+b^{2}>c^{2}\), the triangle is acute.
• If \(a^{2}+b^{2}

Here, the sides are 6, 27, and 29. The longest side \(c = 29\), \(a = 6\), \(b = 27\).

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

First, calculate \(a^{2}\): \(6^{2}=36\)
Then, calculate \(b^{2}\): \(27^{2} = 729\)
So, \(a^{2}+b^{2}=36 + 729=765\)

Now, calculate \(c^{2}\): \(29^{2}=841\)

Step3: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

We have \(765<841\) (since \(a^{2}+b^{2}=765\) and \(c^{2}=841\)). So, by the theorem, the triangle is obtuse.

Answer:

obtuse