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English: Simplified version of similar triangles proof for Pythagoras' theorem. In triangle ACB, angle ACB is the right angle. CH is a perpendicular on hypotenuse AB of triangle ACB. In triangle AHC and triangle ACB, ∠AHC=∠ACB as each is a right angle. ∠HAC=∠CAB as they are common angles at vertex A.
Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles). Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one other side have lengths in the ...
English: Similar triangles proof for Pythagoras' theorem. In triangle ACB, angle ACB is the right angle. CH is a perpendicular on hypotenuse AB of triangle ACB. In triangle AHC and triangle ACB, ∠AHC=∠ACB as each is a right angle. ∠HAC=∠CAB as they are common angles at vertex A. Thus triangle AHC is similar to triangle ACB by AA test.
Area of triangle C = sum of areas of A and B. All three right triangles are similar, so all three areas are proportional to the side bordering the centre triangle. Hence, α(a2 + b2) = α c2, or dividing by α, we have Pythagoras' theorem.
The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. [39] Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same ...
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