The length of the hypotenuse of an isosceles right angled triangle, whose one side is $4\sqrt{2}$ cm, is:

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Accepted Answer

To determine the length of the hypotenuse of an isosceles right-angled triangle, we need to understand the properties of such a triangle. In an isosceles right-angled triangle, the two legs are equal in length, and the hypotenuse is the longest side opposite the right angle.

Given that one side of the triangle is $4\sqrt{2}$ cm, we can deduce that this side is one of the equal legs of the triangle. In an isosceles right-angled triangle, the relationship between the legs and the hypotenuse is given by the Pythagorean theorem:

$$ \text{Hypotenuse} = \sqrt{(\text{Leg})^2 + (\text{Leg})^2} $$

Since both legs are equal, we can simplify this to:

$$ \text{Hypotenuse} = \sqrt{2 \times (\text{Leg})^2} = \sqrt{2} \times \text{Leg} $$

Substituting the given leg length:

$$ \text{Hypotenuse} = \sqrt{2} \times 4\sqrt{2} $$

$$ \text{Hypotenuse} = \sqrt{2} \times 4 \times \sqrt{2} $$

$$ \text{Hypotenuse} = 4 \times 2 $$

$$ \text{Hypotenuse} = 8 \text{ cm} $$

Thus, the length of the hypotenuse is 8 cm, which corresponds to Option 2.

Now, let's explain why the other options are incorrect:

1. **Option 1: 12 cm** - This option is incorrect because, as calculated, the hypotenuse is 8 cm, not 12 cm. The calculation using the properties of an isosceles right-angled triangle does not support this value.

2. **Option 3: $8\sqrt{2}$ cm** - This option is incorrect because it suggests a hypotenuse longer than the calculated 8 cm. The formula for the hypotenuse in an isosceles right-angled triangle does not yield this result when the leg is $4\sqrt{2}$ cm.

3. **Option 4: $12\sqrt{2}$ cm** - This option is incorrect because it significantly overestimates the hypotenuse length. The correct calculation shows that the hypotenuse is 8 cm, not $12\sqrt{2}$ cm.

In conclusion, the correct answer is Option 2, which is 8 cm, based on the properties of an isosceles right-angled triangle and the given side length.