SOLUTION: If triangle ABC is a triangle such that a^2 + b^2 = c^2, then angle BCA is a right triangle? This is the converse to the Pythagorean Theorem. Please do not use law of cosine!

Algebra ->  Algebra  -> Geometry-proofs -> SOLUTION: If triangle ABC is a triangle such that a^2 + b^2 = c^2, then angle BCA is a right triangle? This is the converse to the Pythagorean Theorem. Please do not use law of cosine!      Log On

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Question 192713This question is from textbook
: If triangle ABC is a triangle such that a^2 + b^2 = c^2, then angle BCA is a right triangle? This is the converse to the Pythagorean Theorem. Please do not use law of cosine!This question is from textbook

Answer by jojo14344(1512) About Me  (Show Source):
You can put this solution on YOUR website!


If triangle ABC has dimensions a%5E2%2Bb%5E2=c%5E2, we want to prove angle BCA is a Right Triangle.

Remember a Square is formed by four Right Triangles, being each triangle has the same dimensions as the other triangles.


See figure below:
drawing%28400%2C400%2C-1%2C8%2C-1%2C8%2Cline%280%2C0%2C0%2C6%29%2Cline%280%2C6%2C6%2C6%29%2Cline%286%2C6%2C6%2C0%29%2Cline%286%2C0%2C0%2C0%29%2Cgreen%28line%280%2C2%2C2%2C6%29%29%2Cgreen%28line%282%2C6%2C6%2C4%29%29%2Cgreen%28line%286%2C4%2C4%2C0%29%29%2Cgreen%28line%284%2C0%2C0%2C2%29%29%2Cgreen%28locate%282%2C5.7%2C90%5E0%29%29%2Cgreen%28locate%28.3%2C2.5%2C90%5Eo%29%29%2Cgreen%28locate%283.5%2C.9%2C90%5Eo%29%29%2Cgreen%28locate%285%2C4%2C90%5Eo%29%29%2Cline%280%2C.3%2C.3%2C.3%29%2Cline%28.3%2C.3%2C.3%2C0%29%2Cline%285.7%2C0%2C5.7%2C.3%29%2Cline%285.7%2C.3%2C6%2C.3%29%2Cline%285.7%2C6%2C5.7%2C5.7%29%2Cline%285.7%2C5.7%2C6%2C5.7%29%2Cline%28.3%2C6%2C.3%2C5.7%29%2Cline%28.3%2C5.7%2C0%2C5.7%29%2Clocate%285%2C3%2Cc%29%2Clocate%281.5%2C1.8%2Cc%29%2Clocate%282%2C-.2%2Ca%29%2Clocate%285%2C-.2%2Cb%29%2Clocate%286.2%2C2%2Ca%29%2Clocate%286.2%2C5%2Cb%29%29

We get the Area of the outer Square:
A%5Bo%5D=%28a%2Bb%29%5E2
Also,
A%5Bo%5D=c%5E2%2Bhighlight%284%281%2F2%29%28ab%29%29

*Note: 4%281%2F2%29%28ab%29 ----> Area of 4 Right Triangles outside the inner square.

Therefore,
A%5Bo%5D=A%5Bo%5D
%28a%2Bb%29%5E2=c%5E2%2B%28cross%284%292%281%2Fcross%282%291%29%28ab%29%29
a%5E2%2B2ab%2Bb%5E2=c%5E2%2B2ab
a%5E2%2Bcross%282ab%29%2Bb%5E2=c%5E2%2Bcross%282ab%29
red%28a%5E2%2Bb%5E2=c%5E2%29

Then, Triangle ABC has sides a%5E2%2Bb%5E2=c%5E2

drawing%28200%2C200%2C-1%2C3%2C-1%2C5%2Ctriangle%280%2C0%2C1.5%2C4%2C1.5%2C0%29%2Cline%281.3%2C0%2C1.3%2C.3%29%2Cline%281.3%2C.3%2C1.5%2C.3%29%2Clocate%28.7%2C-.15%2Cb%29%2Clocate%281.7%2C2%2Ca%29%2Clocate%28.7%2C2.9%2Cc%29%2Cred%28locate%281.5%2C4.4%2CB%29%29%2Cred%28locate%281.7%2C.12%2CC%29%29%2Cred%28locate%28-.15%2C0%2CA%29%29%29 ---> Angle BCA is Right Triangle

Thank you,
Jojo