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The Pythagorean Theorem states that the square of the length of the hypotenuse of any right triangle equals the sum of the squares of the lengths of the legs. |
The action figure below gives a wonderful illustration of a geometric proof of the Pythagorean Theorem. |
Experiment with the action figure below by sliding the small blue squares in the left hand figure along the axes. |
Below is the algebraic explanation of how this figure shows that |
In order to talk about the different parts of this figure, let's give some labels to the right-hand square of the action figure above. These labels are shown in Figure 1. |
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Consider this square, ABCD, with side lengths Locate P on side AB so that In the same way, locate Q, R and S on the sides BC, CD and DA respectively.
We see that <APB, <BQC, <CRD and <DSA are straight angles. That is, they measure 180o. We want to show that <SPQ, <PQR, <QRS and <RSP are each right angles in order to show that the interiour quadrilateral, PQRS, is a square.
| (By construction, we already know all four sides of this quadrilateral have the same length, w.) Notice that | ![]() ![]() Thus <APS + <QPB = 90o. | This implies that In the same way we can show that | ![]() ![]() ![]() ![]() Now we compute the area of ABCD in two ways to finally show that |
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Because the area of ABCD computed by either method must be the same, we can equate these two expressions and arrive at:
| ==> u2 + v2 = w2. |
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