- The problem is solved by verifying the triangle inequality \(a + b > c\) and comparing \(a^2 + b^2\) to \(c^2\) for each set.
- A triangle is obtuse if \(a^2 + b^2 < c^2\).
- Testing each set: \((4, 7, 8)\) is acute, \((3, 4, 5)\) is right, \((2, 2, 3)\) is obtuse, \((6, 8, 9)\) is acute, and \((3, 5, 6)\) is obtuse.
- The obtuse sets are \(\boxed{(2, 2, 3), (3, 5, 6)}\).
Explanation
- Analysis of the criteria for classifying triangles.
To determine if a set of three numbers \((a, b, c)\) (where \(c\) is the longest side) represents an obtuse triangle, we must first verify that they form a triangle and then check the relationship between the squares of the sides.
- Triangle Inequality: The sum of the two shorter sides must be strictly greater than the longest side, \(a + b > c\).
- Classification:
- If \(a^2 + b^2 < c^2\), the triangle is obtuse.
- If \(a^2 + b^2 = c^2\), the triangle is a right triangle.
- If \(a^2 + b^2 > c^2\), the triangle is acute.
We will test each given set: \((4, 7, 8)\), \((3, 4, 5)\), \((2, 2, 3)\), \((6, 8, 9)\), and \((3, 5, 6)\).
- Evaluation of the given sets of numbers.
Now we check each candidate set:
- Set \((4, 7, 8)\):
- Triangle inequality: \(4 + 7 = 11 > 8\). (True)
- Squares check: \(4^2 + 7^2 = 16 + 49 = 65\). The square of the longest side is \(8^2 = 64\).
- Since \(65 > 64\), the triangle is acute.
- Set \((3, 4, 5)\):
- Triangle inequality: \(3 + 4 = 7 > 5\). (True)
- Squares check: \(3^2 + 4^2 = 9 + 16 = 25\). The square of the longest side is \(5^2 = 25\).
- Since \(25 = 25\), the triangle is right.
- Set \((2, 2, 3)\):
- Triangle inequality: \(2 + 2 = 4 > 3\). (True)
- Squares check: \(2^2 + 2^2 = 4 + 4 = 8\). The square of the longest side is \(3^2 = 9\).
- Since \(8 < 9\), the triangle is obtuse.
- Set \((6, 8, 9)\):
- Triangle inequality: \(6 + 8 = 14 > 9\). (True)
- Squares check: \(6^2 + 8^2 = 36 + 64 = 100\). The square of the longest side is \(9^2 = 81\).
- Since \(100 > 81\), the triangle is acute.
- Set \((3, 5, 6)\):
- Triangle inequality: \(3 + 5 = 8 > 6\). (True)
- Squares check: \(3^2 + 5^2 = 9 + 25 = 34\). The square of the longest side is \(6^2 = 36\).
- Since \(34 < 36\), the triangle is obtuse.
- Concluding the result.
Based on our analysis, the sets that represent the sides of an obtuse triangle are \((2, 2, 3)\) and \((3, 5, 6)\).
The final answer is: \(\boxed{(2, 2, 3), (3, 5, 6)}\).
Examples
Knowing how to classify triangles by their side lengths is essential in fields like structural engineering and architecture. For example, if you are designing roof trusses, an obtuse triangle might require different bracing than an acute one to handle the stresses effectively. Calculating these properties ensures that the structures remain stable under load, keeping buildings safe and durable.