In a right triangle, one of the acute angles measures 45°, and the side opposite this 45° angle is 8 cm long. What is the length (in centimetres) of the side opposite the right angle (the hypotenuse)?

Introduction / Context:
This question concerns a special right triangle where one of the acute angles is 45 degrees. Such a triangle is known as a 45°-45°-90° right triangle. In these triangles, the legs are equal and the hypotenuse has a fixed relationship with the legs. Using this special ratio, we can easily find the length of the hypotenuse when the length of a leg is known.

Given Data / Assumptions:
- The triangle is a right triangle.- One acute angle is 45°.- The side opposite the 45° angle is 8 cm.- The side opposite the 90° angle is the hypotenuse.- The triangle is a standard 45°-45°-90° triangle, so the other acute angle is also 45°.

Concept / Approach:
In a right triangle with angles 45°, 45°, and 90°, the two legs opposite the 45° angles are equal in length, and the hypotenuse is equal to leg * √2. This ratio arises from applying the Pythagorean theorem to an isosceles right triangle. Therefore, if one leg is known, we can directly multiply it by √2 to obtain the hypotenuse. No complicated trigonometry is needed; the special triangle ratio is sufficient.

Step-by-Step Solution:
Step 1: Recognise that the triangle is a 45°-45°-90° triangle because it is right angled and one acute angle is 45°.Step 2: In such a triangle, the two legs are equal in length.Step 3: We are told that the side opposite the 45° angle is 8 cm, so one leg = 8 cm.Step 4: The hypotenuse h in a 45°-45°-90° triangle is given by h = leg * √2.Step 5: Substitute leg = 8 cm into the formula: h = 8 * √2 cm.Step 6: Therefore, the length of the hypotenuse is 8√2 centimetres.

Verification / Alternative check:
We can verify this by using the Pythagorean theorem. If both legs are of length 8 cm, then h^2 = 8^2 + 8^2 = 64 + 64 = 128. Therefore, h = √128 = √(64 * 2) = 8√2. This matches the result obtained from the special triangle ratio, confirming that 8√2 cm is correct for the hypotenuse of this right triangle.

Why Other Options Are Wrong:
- 4√2 cm: This corresponds to a leg of 4 cm, which contradicts the given leg length of 8 cm.- 8√3 cm: This length would arise in a different triangle, such as a 30°-60°-90° triangle under specific conditions, not in a 45°-45°-90° triangle.- 4√3 cm: This is inconsistent with the Pythagorean relation for legs of length 8 cm.- 16 cm: This is equal to twice the leg length, and 16^2 = 256, which does not equal 8^2 + 8^2 = 128.

Common Pitfalls:
Some students confuse the special triangle ratios and may incorrectly use the 30°-60°-90° triangle ratio instead of the 45°-45°-90° ratio. Others might forget that both legs are equal in a 45°-45°-90° triangle, or they may mistakenly apply the Pythagorean theorem using incorrect side assignments. Remembering that in a 45°-45°-90° triangle, hypotenuse = leg * √2 helps quickly and accurately solve such problems.

Final Answer:
The length of the hypotenuse is 8√2 centimetres.