In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean theorem.

The Pythagorean theorem states that in a right-angled 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 (the legs of the triangle). This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.

Let's break down the explanation further:

1. Right-angled Triangle:
- A right-angled triangle is a triangle that has one angle measuring 90 degrees (a right angle).
- The side opposite the right angle is called the hypotenuse, while the other two sides are called the legs.

2. Hypotenuse:
- The hypotenuse is the longest side of a right-angled triangle.
- It is opposite the right angle.

3. Legs:
- The legs of a right-angled triangle are the two sides that form the right angle.
- One leg is adjacent to the angle, and the other leg is opposite the angle.

4. Pythagorean Theorem:
- The Pythagorean theorem states that in any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
- Mathematically, it can be represented as: c² = a² + b², where c is the length of the hypotenuse, and a and b are the lengths of the legs.

5. Example:
- Let's consider a right-angled triangle with sides of lengths a = 3 and b = 4.
- Using the Pythagorean theorem, we can calculate the length of the hypotenuse (c): c² = 3² + 4² = 9 + 16 = 25.
- Taking the square root of both sides, we find that c = 5.
- Therefore, the length of the hypotenuse in this triangle is 5, which confirms the validity of the Pythagorean theorem.

In conclusion, the correct answer is option 'A' (hypotenuse) because the Pythagorean theorem states that in any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Right-Angled Triangle and Pythagorean Theorem
A right-angled triangle is a triangle that has one angle measuring 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Understanding the Pythagorean Theorem
Let's consider a right-angled triangle with sides a, b, and c, where c is the hypotenuse.

According to the Pythagorean theorem, we can express the relationship between the sides of the triangle as follows:

a^2 + b^2 = c^2

This equation represents a fundamental relationship in geometry that holds true for all right-angled triangles.

Applying the Pythagorean Theorem
In the given question, we are asked to identify which side of the right-angled triangle satisfies the condition that the square of its length is equal to the sum of the squares of the lengths of the other two sides.

Let's consider the options provided:

a) Hypotenuse: The hypotenuse is the side opposite the right angle in a right-angled triangle. According to the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, option A is correct.

b) Altitude: The altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side. The altitude does not satisfy the condition mentioned in the question.

c) Base: The base of a triangle is one of the legs of the right-angled triangle. The base does not satisfy the condition mentioned in the question.

d) None of these: Since option A (hypotenuse) is correct, the answer cannot be "None of these."

Therefore, the correct answer is option A, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.