Asked by Ian
Given is the right triangle ABC with altitude CO. Which one of the following statements is true and contributes to the proof of the Pythagorean Theorem? A. triangle ABC-triangle AOC. B. triangle ABC-triangle BOC. C. Triangle AOC- triangle BOC, D. triangle AOC- triangle COB.
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Answer: A (triangle ABC ~ triangle AOC).
Reason: CO is the altitude to hypotenuse AB, so ∠C = 90° and ∠AOC = 90°, and the two triangles share ∠A. Hence triangle ABC and triangle AOC are similar. (Similarly ABC ~ BOC as well.)
From the similarity ABC ~ AOC we get AC/AB = AO/AC ⇒ AC^2 = AB·AO. From ABC ~ BOC we get BC^2 = AB·BO. Adding these two equalities and using AO + BO = AB gives AB^2 = AC^2 + BC^2, the Pythagorean theorem.
Reason: CO is the altitude to hypotenuse AB, so ∠C = 90° and ∠AOC = 90°, and the two triangles share ∠A. Hence triangle ABC and triangle AOC are similar. (Similarly ABC ~ BOC as well.)
From the similarity ABC ~ AOC we get AC/AB = AO/AC ⇒ AC^2 = AB·AO. From ABC ~ BOC we get BC^2 = AB·BO. Adding these two equalities and using AO + BO = AB gives AB^2 = AC^2 + BC^2, the Pythagorean theorem.
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