272 CHAPTER 8 GEOMETRY FOR AMERICANS

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B D C

We should point out that 8.3.1 and 8.3.2 are intuitively obvious by visualizing

a parallelogram as a "stack" of lines (like a deck of cards, but with infinitely many

"cards" of zero thickness): sliding the stack around, as long as it stays parallel to the

base, shouldn't change the area. The area should equal the base length (which is the

constant cross-sectional length) times the height. In the figure below, the two areas are

equal.

Let's employ this idea-the shearing tool-to prove the most famous theorem in ele­

mentary mathematics.

Example 8.3.5 Prove the Pythagorean Theorem, which states that the sum of the

squares of the legs of a right triangle equals the square of the hypotenuse.

Proof· Let ABC be a right triangle with right angle at B. Then we wish to prove

that

This is something new: an equation involving lengths multiplied by lengths. So far,

only one geometrical concept allows us to think about such things, namely area. So we

are naturally led to the most popular recasting of the Pythagorean Theorem: to show

that the sum of the areas of the two small squares below is equal to the area of the

largest square.