Step-by-step explanation:
A lot of times, one of the hardest parts of proving anything is figuring out exactly what you're trying to prove. Fortunately for us, we already have it laid out. We're trying to prove the Pythagorean theorem, which says that on a right triangle with legs a and b and a hypotenuse c, the sides are related by the equation
[tex]a^2+b^2=c^2[/tex]
This particular proof uses similarity, cleverly splitting the original right triangle into three - the original, and two smaller ones. One fact about similar triangles that helps us is that corresponding sides are proportional. This fact is used for the second step of the proof:
[tex]\frac{c}{b}=\frac{b}{d}[/tex] and [tex]\frac{c}{a}= \frac{a}{f}[/tex]
For our proof, we're going to try to go off of this fact to build back the Pythagorean theorem. We'll need an a², a b², and a c² for that, and we can get the first two by cross multiplying each equations, giving us two new ones:
[tex]cd=b^2[/tex] and [tex]cf=a^2[/tex].
This is Statement 3, and we go to it through Multiplication. Next, we can add the two equations together to get the left side of the Pythagorean theorem, factoring a c out of the right side to give us:
[tex]a^2+b^2=cf+cd=c(f+d)[/tex]
This is Statement 4, and we used the Addition property of equality to get it.
Now, looking at the original figure, we can see that the lengths d and f add together to give us a c. We can express that fact with the equation
[tex]d+f=c[/tex]
This is Statement 5, given to us through the Segment addition postulate. Finally, we can use our equation from statement 5 to substitute f + d for c in our equation from Statement 4, giving us the Pythagorean theorem:
[tex]a^2+b^2=c\cdot c=c^2[/tex]
This is Statement 6, our final statement, and it uses the Substitution property of equality to finish our proof.
