<prompt> Hlalosa hore na ke hobaneng ha Theorem ea Pythagorean e sebetsa.nan  <response>: Instruction in English: Explain why the Pythagorean Theorem works. . Response in English: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse the side opposite the right angle is equal to the sum of the squares of the lengths of the other two sides, or legs. Mathematically, this can be expressed as c2 a2 b2, where c is the hypotenuse and a and b are the legs. This theorem works because it is based on the relationship between the sides of a right triangle.One way to understand why the Pythagorean Theorem works is to look at it geometrically. Imagine a right triangle in which the sides of the right angle are of length a and b, and the hypotenuse is of length c. Now, lets create a square with side length c. On each side of the square, we can draw a right triangle, the same as the original, with the hypotenuse on the side of the square and the legs on the outside. The four triangles inside the square will cover an area of 4 12 ab. In the center, there will be a small square left with side length a-b or b-a depending on which side is longer. This squares area will be a-b2 or b-a2.Adding the areas of the central square and the four triangles, and equating it to the area of the larger square, we have c2 a-b2 or b-a2 4 12 ab. Solving this equation, we get c2 a2 b2. This is the same as the Pythagorean Theorem equation. The theorem can also be proven using algebra or trigonometry, but this geometric proof intuitively shows how the sides of a right triangle are related. . Response in Sesotho: Theorem ea Pythagorean e bolela hore khutlo-tharo e nepahetseng, sekoere sa bolelele ba hypotenuse lehlakore le shebaneng le khutlo e nepahetseng le lekana le kakaretso ea lisekoere tsa bolelele ba mahlakore a mang a mabeli, kapa maoto. Ka lipalo, sena se ka hlalosoa e le c2 a2 b2, moo c e leng hypotenuse le a le b e le maoto. Theorem ena e sebetsa hobane e theilwe hodima kamano pakeng tsa mahlakore a kgutlotharo e nepahetseng. Tsela e ngoe ea ho utloisisa hore na ke hobaneng ha Theorem ea Pythagorean e sebetsa ke ho e sheba ka mokhoa oa geometri. Nahana ka khutlotharo e nepahetseng moo mahlakore a lehlakoreng le letona e leng bolelele ba a le b, me hypotenuse e le bolelele ba c. Joale, ha re theheng lisekoere ka bolelele ba lehlakore c. Ka lehlakoreng le leng le le leng la sekoere, re ka taka khutlotharo e nepahetseng, e tšoanang le ea pele, ka hypotenuse ka lehlakoreng la lisekoere le maoto ka ntle. Likhutlotharo tse nè ka hare ho sekwere li tla koahela sebaka sa 4 12 ab. Bohareng, ho tla ba le sekoere se senyane se setseng se nang le bolelele ba lehlakore ab kapa ba ho latela hore na lehlakore le lelelele hakae. Sebaka sa sekwere sena e tla ba ab2 kapa ba2. Ho eketsa libaka tsa lisekoere tse bohareng le likhutlo tse nè, le ho le lekanya le sebaka sa lisekoere tse kholoanyane, re na le c2 ab2 kapa ba2 4 12 ab. Ho rarolla equation ena, re fumana c2 a2 b2. Sena se tšoana le Pythagorean Theorem equation. Theorem e ka boela ea pakoa ka ho sebelisa aljebra kapa trigonometry, empa bopaki bona ba geometri ka intuitively bo bontša hore na mahlakore a khutlotharo e nepahetseng a amana joang. 