Question

the sides of a triangle have lengths 5, 7, and 9. what kind of triangle is it?
acute right obtuse
submit

Explanation:

Step1: Recall the triangle type rule

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

Step2: Identify the sides

Here, the sides are \(5\), \(7\), and \(9\). The longest side \(c = 9\), and \(a = 5\), \(b = 7\).

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

Calculate \(a^{2}+b^{2}\): \(5^{2}+7^{2}=25 + 49=74\)
Calculate \(c^{2}\): \(9^{2}=81\)

Step4: Compare the values

Since \(74<81\) (i.e., \(a^{2}+b^{2}c^{2}\), acute; \(a^{2}+b^{2}=c^{2}\), right; \(a^{2}+b^{2}

• Acute if \(c^{2}
• Right if \(c^{2}=a^{2}+b^{2}\)
• Obtuse if \(c^{2}>a^{2}+b^{2}\)

Wait, I had the inequality reversed earlier. Let's correct that.

So \(c = 9\), \(a = 5\), \(b = 7\)

\(a^{2}+b^{2}=5^{2}+7^{2}=25 + 49 = 74\)

\(c^{2}=9^{2}=81\)

Since \(81>74\) (i.e., \(c^{2}>a^{2}+b^{2}\)), the triangle is obtuse? Wait, no, now I am confused. Wait, let's take an example. For a right triangle, \(3\), \(4\), \(5\): \(3^{2}+4^{2}=9 + 16 = 25=5^{2}\). For an acute triangle, say \(2\), \(3\), \(4\): \(2^{2}+3^{2}=4 + 9 = 13\), \(4^{2}=16\), \(13<16\), so it is obtuse? Wait, no, \(2\), \(3\), \(4\): the angles. Let's calculate the angle opposite the side \(4\) using the Law of Cosines. \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}=\frac{4 + 9-16}{2\times2\times3}=\frac{-3}{12}=-\frac{1}{4}\), which is negative, so the angle is obtuse. So yes, if \(c^{2}>a^{2}+b^{2}\), the angle opposite \(c\) is obtuse, so the triangle is obtuse. Wait, but in our problem, \(5\), \(7\), \(9\): \(c = 9\), \(a = 5\), \(b = 7\). \(a^{2}+b^{2}=25 + 49 = 74\), \(c^{2}=81\). Since \(81>74\), the triangle is obtuse. But wait, the initial mistake was in the direction of the inequality. So the correct answer is obtuse? Wait, no, wait I think I messed up the rule. Let me check again. The rule is:

For a triangle with sides \(a\), \(b\), \(c\) ( \(c\) is the longest side):

• If \(c^{2}=a^{2}+b^{2}\), right triangle.
• If \(c^{2}
• If \(c^{2}>a^{2}+b^{2}\), obtuse triangle (because the square of the longest side is greater than the sum of the squares of the other two, so the angle opposite the longest side is obtuse).

So in our case, \(c = 9\), \(a = 5\), \(b = 7\). \(c^{2}=81\), \(a^{2}+b^{2}=74\). Since \(81>74\), the triangle is obtuse. So the correct answer is obtuse.

Answer:

acute