Question

the sides of a triangle have lengths 14, 39, and 42. what kind of triangle is it?
acute
right
obtuse

Explanation:

Step1: Recall the triangle inequality theorem for types

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), we use the Pythagorean inequality:

• If \(a^{2}+b^{2}=c^{2}\), it's a right triangle.
• If \(a^{2}+b^{2}>c^{2}\), it's an acute triangle.
• If \(a^{2}+b^{2}

Here, \(a = 14\), \(b = 39\), \(c = 42\) (since 42 is the longest side).

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

First, calculate \(a^{2}\): \(14^{2}=196\)
Then, calculate \(b^{2}\): \(39^{2}=1521\)
Sum them: \(a^{2}+b^{2}=196 + 1521=1717\)
Now, calculate \(c^{2}\): \(42^{2}=1764\)

Step3: Compare the two results

We see that \(1717<1764\), which means \(a^{2}+b^{2}

Answer:

obtuse