Final answer:
The lengths of the unknown sides of a right-angled triangle can be determined using the Pythagorean theorem if two side lengths are known. Trigonometric ratios can be used if one side length and one non-right angle are known. Without sufficient information, the lengths cannot be calculated.
Step-by-step explanation:
To determine the lengths of the unknown sides of a right-angled triangle, it is essential to apply geometric principles such as the Pythagorean theorem. If we know the lengths of two sides of a right-angled triangle, we can calculate the third side using the Pythagorean theorem, which states that a² + b² = c², where 'c' represents the hypotenuse and 'a' and 'b' are the lengths of the other two sides.
For instance, if the lengths of the legs are 'a' and 'b', the hypotenuse 'c' can be determined by taking the square root of the sum of the squares of the other two sides, which is written as c = √a² + b². Without sufficient information, such as one side length and at least another side length or an angle, we cannot determine the lengths of the other sides. Ratios and proportions can also apply if the triangles are similar, in which case the corresponding sides of the triangles would be in proportion with each other.
When considering measurement units, trigonometric ratios are unitless and can be used to relate the angles and sides of right-angled triangles. For example, if you know one angle (other than the right angle) and one side length, you can use trigonometric functions (sine, cosine, or tangent) to find the unknown side lengths.