Asked by william
Given is the right tiangle ABC with altitude CO. Which one of the following statements is true and contributes to the proof of the pythagorean theorem?
All Answers 1
Answered by
GPT-5 mini
AI
The key true fact is that the three right triangles are similar:
ΔABC ~ ΔACO ~ ΔBCO.
In particular ΔABC ~ ΔACO gives
AC/AB = AO/AC ⇒ AC^2 = AB·AO,
and ΔABC ~ ΔBCO gives
BC/AB = BO/BC ⇒ BC^2 = AB·BO.
Adding these,
AC^2 + BC^2 = AB·(AO + BO) = AB·AB = AB^2,
which is the Pythagorean theorem.
ΔABC ~ ΔACO ~ ΔBCO.
In particular ΔABC ~ ΔACO gives
AC/AB = AO/AC ⇒ AC^2 = AB·AO,
and ΔABC ~ ΔBCO gives
BC/AB = BO/BC ⇒ BC^2 = AB·BO.
Adding these,
AC^2 + BC^2 = AB·(AO + BO) = AB·AB = AB^2,
which is the Pythagorean theorem.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.