Pythagorean Theorem

Pythagorean Theorem

      The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Proof:

Let a, b, and c be the lengths of the sides of a right triangle, with c being the length of the hypotenuse. We want to show that:

c^2 = a^2 + b^2

We can start by drawing a square with side length c, as shown in the diagram below:

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We can see that the total area of the larger square with side length c is equal to the sum of the areas of the smaller squares with side lengths a and b. Therefore, we have:

c^2 = a^2 + b^2

This completes the proof.

Additional Proofs:

Algebraic Proof:

We can also prove the Pythagorean Theorem using algebra. Starting with the equation:

c^2 = a^2 + b^2

We can rearrange it as:

c^2 - a^2 = b^2

We can then factor the left-hand side using the difference of squares formula:

(c - a)(c + a) = b^2

We can then take the square root of both sides:

c - a = sqrt(b^2) = b

or

c + a = sqrt(b^2) = b

Solving for c in each equation, we get:

c = a + b or c = b - a

Since a, b, and c are all positive, we must have:

c = a + b

Therefore, we have:

c^2 = (a + b)^2 = a^2 + 2ab + b^2

But we also have:

c^2 = a^2 + b^2

Therefore, we must have:

a^2 + 2ab + b^2 = a^2 + b^2

Simplifying this equation, we get:

2ab = 0

Since a and b are both positive, we must have:

a = 0 or b = 0

But this contradicts the fact that a, b, and c are the lengths of the sides of a right triangle, so we must have:

c^2 = a^2 + b^2

Geometric Proof:

Another way to prove the Pythagorean Theorem is to use similar triangles. Starting with a right triangle with sides a, b, and c

This divides the larger triangle into two smaller triangles, both of which are similar to the original triangle. Using the notation in the diagram, we can write:

a/b = e/c

b/d = c/f

Multiplying these equations, we get:

(a/b) * (b/d) = (e/c) * (c/f)

Simplifying, we get:

a/d = e/f

Since the triangles are similar, we also have:

a + e = c

b + d = c

e + f = b

Substituting these equations into the previous equation, we get:

a/d = (c - b)/(b - d)

Simplifying, we get:

a(d - b) = c(b - d)

Expanding both sides and simplifying, we get:

a^2 + b^2 = c^2

This completes the proof using similar triangles.

Trigonometric Proof:

Multiplying these equations, we get:

(a/b) * (b/d) = (e/c) * (c/f)

Simplifying, we get:

a/d = e/f

Let θ be the angle opposite side a, and let h be the length of the altitude from the right angle to side c. 

Using trigonometry, we have:

sin θ = a/c

cos θ = b/c

tan θ = a/b

Squaring the second and third equations and adding them, we get:

cos^2 θ + sin^2 θ = a^2/b^2 + 1

Multiplying both sides by b^2 and simplifying, we get:

b^2 = a^2 + c^2

Substituting h for c (since h is the length of the altitude from the right angle to side c), we get:

b^2 = a^2 + h^2

This completes the proof using trigonometry.

Overall, the Pythagorean Theorem is a fundamental result in geometry that has been proven using a variety of different methods.