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

Labels:
Einstein,
Problem 264,
Pythagoras,
right triangle,
similarity,
square

Subscribe to:
Post Comments (Atom)

See the

ReplyDeletedrawingDefine 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