If the sides of the triangle are 12, 35 and 37 ,
determine whether the triangle is a right
angled triangle . Justify.

Answer

To determine whether the triangle with sides 12, 35, and 37 is a right-angled triangle, we will use the Pythagorean theorem. According to this theorem, for a triangle to be a right-angled triangle, the square of the length of the hypotenuse (the longest side) must equal the sum of the squares of the lengths of the other two sides.

### Step-by-step solution:

1. **Identify the sides of the triangle**:
The given sides are 12, 35, and 37.
Here, the longest side is 37, so we will consider it as the hypotenuse (c).

2. **Assign values to the sides**:
Let:
a = 12 (one side)
b = 35 (another side)
c = 37 (hypotenuse)

3. **Apply the Pythagorean theorem**:
According to the theorem:
\( c^2 = a^2 + b^2 \)
We need to check if:
\( 37^2 = 12^2 + 35^2 \)

4. **Calculate \( c^2 \)**:
\( 37^2 = 1369 \)

5. **Calculate \( a^2 \)**:
\( 12^2 = 144 \)

6. **Calculate \( b^2 \)**:
\( 35^2 = 1225 \)

7. **Add \( a^2 \) and \( b^2 \)**:
\( 12^2 + 35^2 = 144 + 1225 = 1369 \)

8. **Compare \( c^2 \) with \( a^2 + b^2 \)**:
We have:
\( 37^2 = 1369 \)
\( 12^2 + 35^2 = 1369 \)
Since both sides are equal, we conclude that:
\( 37^2 = 12^2 + 35^2 \)

9. **Conclusion**:
Since the equation holds true, the triangle with sides 12, 35, and 37 is indeed a right-angled triangle.