Question

the sides of a triangle have lengths 10, 15, and 17. what kind of triangle is it?
acute
right
obtuse

Explanation:

Step1: Recall the triangle inequality theorem for angles

For a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), we use the Pythagorean theorem to determine the type of triangle. If \(a^{2}+b^{2}=c^{2}\), it is a right triangle. If \(a^{2}+b^{2}>c^{2}\), it is an acute triangle. If \(a^{2}+b^{2}

Step2: Identify the longest side and calculate the squares

Here, the side lengths are \(10\), \(15\), and \(17\). The longest side \(c = 17\). Calculate \(a^{2}+b^{2}\) where \(a = 10\) and \(b=15\):
\(a^{2}=10^{2} = 100\)
\(b^{2}=15^{2}=225\)
\(a^{2}+b^{2}=100 + 225=325\)
Calculate \(c^{2}\):
\(c^{2}=17^{2}=289\)

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

Since \(325>289\) (i.e., \(a^{2}+b^{2}>c^{2}\)), the triangle is acute.

Answer:

acute