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 ->  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


   

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(1513) 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:


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

---> Angle BCA is Right Triangle

Thank you,
Jojo