State precisely what’s wrong with the following purported proof of the Pythagorean Theorem Proof. Consider an arbitrary right triangle. Let the two legs and hypotenuse, respectively, have length a, b,...




State precisely what’s wrong with the following purported proof of the Pythagorean Theorem


Proof. Consider an arbitrary right triangle. Let the two legs and hypotenuse, respectively, have length a, b, and c, and let the angles between the legs and the hypotenuse be given by θ and φ = 90◦ − θ. (See Figure 4.35(a).) Draw a line perpendicular to the hypotenuse to the opposite vertex, dividing the interior of the triangle into two separate sections, which are shaded with different colors in Figure 4.35(b). Observe that the unlabeled angle within the smaller shaded interior triangle must be φ = 90◦ − θ, because the other angles of the smaller shaded interior triangle are (just like for the enclosing triangle) 90◦ and θ. Similarly, the unlabeled angle within the larger shaded interior triangle must be θ. Therefore we have three similar triangles, all with angles 90 ◦ , θ, and φ. Call the lengths of the previously unnamed sides x, y, and z as in Figure 4.35(c). Now we can assemble our known facts. By assumption













May 07, 2022
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