Final answer:
None of the options provided directly prove the Pythagorean theorem. Instead, to prove that BC² = AB² + AC² for a right-angled triangle ABC, we must show that triangles ABD and ADC (using an altitude AD) are similar to triangle ABC and then use these relationships to derive the theorem.
This correct answer is none of the above.
Step-by-step explanation:
The task involves proving the Pythagorean theorem for a right-angled triangle ABC with angle A being 90°. According to the theorem, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms, this is written as BC² = AB² + AC², where BC is the hypotenuse.
However, none of the options provided are correct because they all introduce an unrelated term involving DC, which is not part of proving the Pythagorean theorem directly. To use segment AD in the proof, as is indicated in the question, we would normally use the method where triangles ABD and ADC are similar to triangle ABC, which is not mentioned in the options.
To prove the Pythagorean theorem, we would say that in triangle ABC, with AD as the altitude to the hypotenuse BC, the triangles ABD and ADC are similar to ABC. Therefore, we can set up the following two equations using similarity:
AB²/AD² = BC²/AB² (for triangle ABD similar to ABC)
and
AC²/AD² = BC²/AC² (for triangle ADC similar to ABC).
Adding these two ratios together and solving for BC², we get the Pythagorean theorem:
BC² = AB² + AC².
This correct answer is none of the above.