#### Question

Proofs of The Pythagorean Theorem

Proofs of The Pythagorean Theorem

Proof By Area method

Draw three squares using the sides of the right triangle ABC as one of their sides, respectively.

The altitude from the right angle ∠ACB intersects the hypotenuse AB at point J, and the side HI of square ABHI at point K.

Connect point A to G and point C to H

Look at △ABG and △CBH ,

∠ABG =∠CBH = 90° + ∠ABC\kappa

According to the condition of triangles congruent SAS,

△ABG \cong △CBH

In the right trapezoid ABCF,

S_{CBGF} = 2S_{△ABG}

In the right trapezoid CBHK,

S_{BHKJ} = 2S_{△CBH}

\therefore S_{CBGF} =S_{BHKJ}

Similarly,

S_{ACED} = S_{AJKI}

\therefore

S_{CBGF} + S_{ACED} = S_{BHKJ} + S_{AJKI} = S_{ABHI}

\therefore a^2+b^2 = c^2