If in a triangle, square of one side is equal to the sum of squares of the other two sides, then the angle opposite the first side is :

Options

A. `60^@`
B. `90^@`
C. `45^@`
D. `30^@`

Answer

To solve the problem, we need to determine the angle opposite to a side in a triangle when the square of that side is equal to the sum of the squares of the other two sides. This is a direct application of the Pythagorean theorem.

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We are given that in a triangle, the square of one side (let's call it \( a \)) is equal to the sum of the squares of the other two sides (let's call them \( b \) and \( c \)). Mathematically, this can be expressed as:
\[ a^2 = b^2 + c^2 \]

2. **Identify the Triangle**:
Let's denote the triangle as \( \triangle ABC \) where:
- Side \( a \) is opposite angle \( A \)
- Side \( b \) is opposite angle \( B \)
- Side \( c \) is opposite angle \( C \)

3. **Apply the Pythagorean Theorem**:
The equation \( a^2 = b^2 + c^2 \) is the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

4. **Determine the Angle**:
Since \( a^2 = b^2 + c^2 \), it indicates that angle \( A \) (the angle opposite side \( a \)) must be a right angle. Therefore:
\[ \angle A = 90^\circ \]

5. **Conclusion**:
Thus, the angle opposite the first side (side \( a \)) is:
\[ \text{Angle opposite the first side } = 90^\circ \]

### Final Answer:
The angle opposite the first side is \( 90^\circ \).

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