Question

determine if the following measures will form a right, acute, or obtuse triangle, or no triangle at all.
6, 11, 14 select
18, 24, 30 select
21, 13, 24 select
10, 8, 2 select

Explanation:

To determine the type of triangle (or if no triangle is formed) from side lengths \(a\), \(b\), \(c\) (where \(c\) is the longest side), we use the triangle inequality theorem (to check if a triangle is formed) and the Pythagorean theorem's converse:

• Triangle Inequality: \(a + b > c\) (for the longest side \(c\)) must hold.
• Pythagorean Converse:
• If \(a^2 + b^2 = c^2\), right triangle.
• If \(a^2 + b^2 > c^2\), acute triangle.
• If \(a^2 + b^2 < c^2\), obtuse triangle.

For \(6, 11, 14\):

Step 1: Check Triangle Inequality

Longest side \(c = 14\). Check \(6 + 11 > 14\): \(17 > 14\) (true, so triangle is formed).

Step 2: Apply Pythagorean Converse

Calculate \(6^2 + 11^2 = 36 + 121 = 157\) and \(14^2 = 196\).
Since \(157 < 196\) (\(a^2 + b^2 < c^2\)), the triangle is obtuse.

For \(18, 24, 30\):

Step 1: Check Triangle Inequality

Longest side \(c = 30\). Check \(18 + 24 > 30\): \(42 > 30\) (true, triangle formed).

Step 2: Apply Pythagorean Converse

Calculate \(18^2 + 24^2 = 324 + 576 = 900\) and \(30^2 = 900\).
Since \(900 = 900\) (\(a^2 + b^2 = c^2\)), the triangle is right.

For \(21, 13, 24\):

Step 1: Check Triangle Inequality

Longest side \(c = 24\). Check \(21 + 13 > 24\): \(34 > 24\) (true, triangle formed).

Step 2: Apply Pythagorean Converse

Calculate \(13^2 + 21^2 = 169 + 441 = 610\) and \(24^2 = 576\).
Since \(610 > 576\) (\(a^2 + b^2 > c^2\)), the triangle is acute.

For \(10, 8, 2\):

Step 1: Check Triangle Inequality

Longest side \(c = 10\). Check \(8 + 2 > 10\): \(10 > 10\) (false, \(10 = 10\)).
Thus, no triangle is formed (since the sum of the two shorter sides is not greater than the longest side).

Answer:

• \(6, 11, 14\): Obtuse Triangle
• \(18, 24, 30\): Right Triangle
• \(21, 13, 24\): Acute Triangle
• \(10, 8, 2\): No Triangle