Classify the triangle as a type of triangle if the sides of the triangle are 6, 12 and 13 units.

Options

A. Right angled triangle
B. Obtuse angled triangle
C. Acute angled triangle
D. None of these

Answer

To classify the triangle with sides of 6, 12, and 13 units, we can follow these steps:

### Step 1: Identify the sides
Let's denote the sides of the triangle as:
- \( a = 6 \) units
- \( b = 12 \) units
- \( c = 13 \) units (the largest side)

### Step 2: Apply the Pythagorean theorem
To classify the triangle, we will use the Pythagorean theorem, which states:
- For a right triangle: \( c^2 = a^2 + b^2 \)
- For an obtuse triangle: \( c^2 > a^2 + b^2 \)
- For an acute triangle: \( c^2 < a^2 + b^2 \)

### Step 3: Calculate \( c^2 \) and \( a^2 + b^2 \)
Now, we will calculate:
- \( c^2 = 13^2 = 169 \)
- \( a^2 = 6^2 = 36 \)
- \( b^2 = 12^2 = 144 \)

Now, add \( a^2 \) and \( b^2 \):
- \( a^2 + b^2 = 36 + 144 = 180 \)

### Step 4: Compare \( c^2 \) with \( a^2 + b^2 \)
Now we compare:
- \( c^2 = 169 \)
- \( a^2 + b^2 = 180 \)

Since \( 169 < 180 \), we conclude that:
- \( c^2 < a^2 + b^2 \)

### Step 5: Classify the triangle
Since \( c^2 < a^2 + b^2 \), the triangle is classified as an **acute triangle**.

### Final Answer
The triangle with sides 6, 12, and 13 units is an **acute triangle**.

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