Question

match the following side lengths with the type of triangle they create. 3, 5, 9; 3, 5, 4.5; 4, 4, 7; 5, 12, 13. options: acute triangle, not a triangle, obtuse triangle, right triangle

Explanation:

To determine the type of triangle (or if it's not a triangle) from side lengths, we use the triangle inequality theorem (for "Not a triangle") and the Pythagorean theorem (or its extensions for acute/obtuse/right triangles). The triangle inequality states that the sum of any two sides must be greater than the third side. For right triangles, \(a^2 + b^2 = c^2\) (where \(c\) is the longest side). For acute triangles, \(a^2 + b^2 > c^2\), and for obtuse triangles, \(a^2 + b^2 < c^2\).

Step 1: Analyze \(3, 5, 9\)

Check triangle inequality: \(3 + 5 = 8\), and \(8 < 9\). So, it does not satisfy the triangle inequality.
Conclusion: Not a triangle.

Step 2: Analyze \(3, 5, 4.5\)

First, check triangle inequality: \(3 + 5 = 8 > 4.5\), \(3 + 4.5 = 7.5 > 5\), \(5 + 4.5 = 9.5 > 3\). Now, check the Pythagorean theorem (longest side is \(5\)): \(3^2 + 4.5^2 = 9 + 20.25 = 29.25\), and \(5^2 = 25\). Since \(29.25 > 25\) (\(a^2 + b^2 > c^2\)), it's an acute triangle.
Conclusion: Acute triangle.

Step 3: Analyze \(4, 4, 7\)

Check triangle inequality: \(4 + 4 = 8 > 7\), \(4 + 7 = 11 > 4\), \(4 + 7 = 11 > 4\). Longest side is \(7\). Calculate \(4^2 + 4^2 = 16 + 16 = 32\), and \(7^2 = 49\). Since \(32 < 49\) (\(a^2 + b^2 < c^2\)), it's an obtuse triangle.
Conclusion: Obtuse triangle.

Step 4: Analyze \(5, 12, 13\)

Check the Pythagorean theorem: \(5^2 + 12^2 = 25 + 144 = 169\), and \(13^2 = 169\). So, \(5^2 + 12^2 = 13^2\).
Conclusion: Right triangle.

Answer:

• \(3, 5, 9\) → Not a triangle
• \(3, 5, 4.5\) → Acute triangle
• \(4, 4, 7\) → Obtuse triangle
• \(5, 12, 13\) → Right triangle