For a right-angled triangle, two small sides are of 6 cm and 8 cm length. Length of third side will be

Question

Accepted Answer

To determine the length of the third side of a right-angled triangle when the lengths of the two smaller sides are given, we use 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.

Given:
- One side (a) = 6 cm
- Another side (b) = 8 cm

We need to find the hypotenuse (c).

According to the Pythagorean theorem:
$$ c^2 = a^2 + b^2 $$

Substitute the given values:
$$ c^2 = 6^2 + 8^2 $$
$$ c^2 = 36 + 64 $$
$$ c^2 = 100 $$

To find the length of the hypotenuse (c), take the square root of both sides:
$$ c = \sqrt{100} $$
$$ c = 10 $$

Therefore, the length of the third side, which is the hypotenuse, is 10 cm. This corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (14 cm):** If the hypotenuse were 14 cm, then $ 14^2 = 196 $. However, the sum of the squares of the other two sides is 100, not 196. Therefore, 14 cm cannot be the correct length of the hypotenuse.

2. **Option 2 (9 cm):** If the hypotenuse were 9 cm, then $ 9^2 = 81 $. Again, the sum of the squares of the other two sides is 100, not 81. Thus, 9 cm is not the correct length of the hypotenuse.

3. **Option 4 (100 cm):** If the hypotenuse were 100 cm, then $ 100^2 = 10000 $. This is far greater than the sum of the squares of the other two sides, which is 100. Therefore, 100 cm is not a feasible length for the hypotenuse.

In conclusion, the only option that satisfies the Pythagorean theorem for the given side lengths is Option 3, which is 10 cm.