In Euclidean geometry, a triangle with sides 5, 12, and 14 is

(a) acute

(b) right

(c) obtuse

To determine the nature of a triangle with sides 5, 12, and 14, we can use the Pythagorean inequality:

1. If \( c^2 < a^2 + b^2 \), the triangle is **acute**.
2. If \( c^2 = a^2 + b^2 \), the triangle is **right**.
3. If \( c^2 > a^2 + b^2 \), the triangle is **obtuse**.

Here, the longest side is 14 (which we'll treat as \( c \)), and the other two sides are 5 and 12 (which we'll treat as \( a \) and \( b \)).

### Step-by-Step Solution

1. Calculate \( a^2 + b^2 \):
   \[
   a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169
   \]

2. Calculate \( c^2 \):
   \[
   c^2 = 14^2 = 196
   \]

3. Compare \( c^2 \) and \( a^2 + b^2 \):
   \[
   196 > 169
   \]

Since \( c^2 > a^2 + b^2 \), the triangle is **obtuse**.

### Answer
The correct choice is:
**(c) obtuse**

Would you like more details on this solution or have any questions?

Here are some related questions to explore:

1. How do we determine if a triangle is acute, right, or obtuse when given only angles?
2. Can a triangle with any arbitrary side lengths form a valid triangle in Euclidean geometry?
3. How would the classification change if we altered one of the side lengths slightly?
4. What is the significance of the Pythagorean Theorem in classifying triangles?
5. Are there other methods besides the Pythagorean inequality to classify triangles by angle type?

**Tip:** Remember that for any three side lengths to form a triangle, they must satisfy the triangle inequality.

Discover how to determine if a triangle with sides 5, 12, and 14 is acute, right, or obtuse in Euclidean geometry. Using the Pythagorean inequality, we classify the triangle's type based on angle properties. Ideal for students in Grades 8-10 learning triangle classification and theorems.

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