<prompt> Hlalosa Theorem ea Pythagorean.nan  <response>: Instruction in English: Prove the Pythagorean Theorem. . Response in English: The Pythagorean Theorem is a fundamental relation between the three sides of a right triangle. It states that the square of the length of the hypotenuse the longest side opposite the right angle is equal to the sum of the squares of the lengths of the other two sides.The theorem can be proved in many ways. One of the simplest proofs is the one attributed to the ancient Greek mathematician Euclid. It uses the concept of similarity of triangles.Consider a right triangle ABC with the right angle at C. Draw a perpendicular line from C to the hypotenuse AB, and let D be the point of intersection.This creates two smaller right triangles, ACD and CBD. These two triangles are similar to the original triangle ABC, as well as to each other.From the similarity of the triangles, we can write the following proportions:ADAC ACAB - AD AB AC2BDBC BCAB - BD AB BC2Adding these two equations, we get:AD BD AB AC2 BC2But, AD BD AB, so:AB2 AC2 BC2This completes the proof of the Pythagorean Theorem. . Response in Sesotho: Theorem ea Pythagorean ke kamano ea mantlha pakeng tsa mahlakore a mararo a khutlo-tharo e nepahetseng. E bolela hore sekoere sa bolelele ba hypotenuse lehlakore le lelelele ka ho fetisisa le shebaneng le lehlakoreng le letona le lekana le kakaretso ea lisekoere tsa bolelele ba mahlakore a mang a mabeli. Theorem e ka pakoa ka litsela tse ngata. Bopaki bo bong bo bonolo ka ho fetisisa ke boo ho thoeng bo ngotsoe ke setsebi sa lipalo sa Mogerike Euclid. E sebelisa mohopolo oa ho tšoana ha likhutlotharo. Nahana ka khutlotharo e nepahetseng ea ABC e nang le angle e nepahetseng ho C. Thala mola oa perpendicular ho tloha ho C ho ea ho hypotenuse AB, me D e be ntlha ea mateano. Sena se theha likhutlo tse peli tse nyane tse nepahetseng, ACD le CBD. Likhutlotharo tsena tse peli li tšoana le khutlotharo ea mantlha ABC, hammoho le tse ling. Ho tsoa ho tšoana ha likhutlo-tharo, re ka ngola likarolo tse latelang: ADAC ACAB - AD AB AC2 BDBC BCAB - BD AB BC2 Ho kenyelletsa lipalo tsena tse peli , re fumana: AD BD AB AC2 BC2 Empa, AD BD AB, kahoo: AB2 AC2 BC2 Sena se phethela bopaki ba Theorem ea Pythagorean. 