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A baseball diamond is really a square measuring 90 feet on a side. How far does the catcher have to throw the ball to get it from homeplate to second base?
How many examples of whole-number side lengths of right triangles can you find? Is it worth remembering these examples? Is there a formula to generate more such triples? All such triples?
Use the four right triangles in and the small square from your manipulatives kit. Create a single large square with side length equal to the hypotenuse. What is the area of this large square? Now rearrange the shapes into a “fat L” shape whose area is comprised of two squares with areas a^{2} and b^{2}.
For many interesting, even animated proofs of the Pythagorean Theorem, see roofs of the Pythagorean Theorem
Is it legitimate to make a geometric proof of an algebraic statement? Is the Pythagorean Theorem an algebraic statement, or a geometric statement?
This section is adapted from: Instructor Resources and Adjunct Guide for the second edition of The Heart of Mathematics by E. Burger, M. Starbird, and D. Bergstrand.
The backstay is the hypotenuse of a right triangle formed by the deck, the mast and the cable. The base, or leg, of the triangle along the deck has length 50 feet. Therefore, if the mast length is m feet, then 130^{2} = 50^{2} + m^{2}. Solving for m, we find that m = 120 feet. Note that the sides of this triangle are in the ratio 5 : 12 : 13 which is very common in these types of problems.
Using some scrap lumber, (or even some good lumber), carefully measure and create a triangle whose sides are in the ratio 3 : 4 : 5. Since this is patio, we could even create a fairly large triangle, whose dimensions are 3 feet, 4 feet, and hypotenuse 5 feet. Since the sides satisfy the Pythagorean relation 3^{2} + 4^{2} = 5^{2}, by the converse to the Pythagorean Theorem, we know it must be a right triangle, with a right angle between the two shorter sides. You can now use this triangle as homemade carpenter’s square to check right angles on your patio. Many scientists believed the ancient Egyptians and Babylonians used a similar idea with knotted ropes for the construction of buildings and monuments.
The area of the smaller, inner circular disk is . The area of the larger, outer circular disk is . Therefore, the area of the doughnut shaped region inside the outer disk but outside the inner disk is However, if we let be the length of the segment that goes from the “North pole” of the inner circle horizontally to the each of the outer circle, then by the Pythagorean theorem, . Rearranging, and so the area of the doughnut shape is . In the second figure, a radius of the shaded medium circle is of length . Therefore the area of the shaded circle is also. The two areas are equal.
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