Proposed Problem
See complete Problem 265 at:
gogeometry.com/problem/p265_pythagorean_theorem_right_triangle.htm
Level: High School, SAT Prep, College geometry
Thursday, March 5, 2009
Problem 265: Right Triangle, Pythagorean Theorem
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See the drawing
ReplyDeleteDefine a square ABCD
Define E in AB, AE=a and EB=b
Define F in BC such as BF=b and FC=a
[ABCD]=[HIGD]+[EBFI]+[AEIH]+[FCGI]
[AEIH]=[FCGI]=ab=>[ABCD]=a^2+b^2+2ab=(a+b)^2
Cut [AEIH] in 2 triangles T1=AEI and T2=AHI
Cut [FCGI] in 2 triangles T3=FCG and T4=FIG
AEI, AHI, FCG and FIG are congruent with surface T=ab/2
Define c=AI=FG
[ABCD]=a^2+b^2+4T
Translate T2 such as HI becomes DG
Translate T1 such as EI becomes BF
Translate T4 such AS GI becomes A’A
By symetry, A’F’FG is a square => [A’F’FG]=c^2
[ABCD] =4T+[A’F’FG] =4T+c^2
But [ABCD]=a^2+b^2+4T
Therefore a^2+b^2=c^2