Three sides of a triangle are 6 cm , 8 cm and 9 cm. The triangle is

Options

A. Acute - angled
B. Right -angled
C. Obtuse-angled
D. Can't be determined

Answer

To determine the type of triangle formed by the sides 6 cm, 8 cm, and 9 cm, we can follow these steps:

### Step 1: Identify the sides
We have three sides of the triangle:
- Side a = 6 cm
- Side b = 8 cm
- Side c = 9 cm

### Step 2: Determine the longest side
The longest side is considered the hypotenuse when checking for right, acute, or obtuse triangles. Here, the longest side is:
- Hypotenuse (c) = 9 cm

### Step 3: Apply the Pythagorean theorem
To classify the triangle, we will use the Pythagorean theorem:
- 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 4: Calculate the squares of the sides
Now we will calculate the squares of the sides:
- \( c^2 = 9^2 = 81 \)
- \( a^2 = 6^2 = 36 \)
- \( b^2 = 8^2 = 64 \)

### Step 5: Sum the squares of the shorter sides
Now we will sum the squares of the two shorter sides:
- \( a^2 + b^2 = 36 + 64 = 100 \)

### Step 6: Compare \( c^2 \) with \( a^2 + b^2 \)
Now we will compare \( c^2 \) with \( a^2 + b^2 \):
- \( c^2 = 81 \)
- \( a^2 + b^2 = 100 \)

Since \( 81 < 100 \), we have:
- \( c^2 < a^2 + b^2 \)

### Step 7: Conclusion
Since \( c^2 < a^2 + b^2 \), the triangle is an **acute triangle**.