Question

the sides of a triangle have lengths 3, 16, and 16. what kind of triangle is it?
acute
right
obtuse

Explanation:

Step1: Recall the triangle type rules

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

• 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, the sides are \(3\), \(16\), \(16\). The longest side \(c = 16\), and the other two sides \(a = 3\), \(b = 16\).

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

First, calculate \(a^{2}+b^{2}\):
\(a^{2}=3^{2}=9\), \(b^{2}=16^{2}=256\), so \(a^{2}+b^{2}=9 + 256=265\)
Then, calculate \(c^{2}\):
\(c^{2}=16^{2}=256\)

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

Since \(265>256\) (i.e., \(a^{2}+b^{2}>c^{2}\)), the triangle is acute.

Answer:

acute