Final answer:
The length of DF in similar right-angled triangles ABC and DEF can be determined using the ratios from the properties of similar triangles and the Pythagorean theorem.
Step-by-step explanation:
To find the length of DF in similar right-angled triangles ABC and DEF, we need to utilize the properties of similar triangles and the Pythagorean theorem. Since the triangles are similar, the ratios of corresponding sides are equal. This implies that AB/DE = AC/DF = BC/EF. We know the Pythagorean theorem states that 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.
If you are given the lengths of any two sides of the triangles, you can set up a proportion based on the similarity to find the unknown sides. For example, if AB, AC, and DE are known, you can find DF by solving the proportion AB/DE = AC/DF. If the length of the hypotenuse and one side are given in one triangle, apply the Pythagorean theorem, for example, AB2 = AC2 + BC2, to find the missing side length.