The Pythagorean Theorem

The most famous theorem in Euclidean geometry is usually credited to Pythagoras (ca. 500 B.C.). However, Babylonian tablets suggest that the result was known more than a thousand years earlier. The theorem states that the square of the hypotenuse c of a right triangle is equal to the sum of the squares of the lengths a and b of the other two sides  

FIGURE 1.1 Geometrical interpretation of Pythagoras theorem. The area of the large square equals the sum of the areas of the two smaller squares. 

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The polar coordinate system describes the location of a point (denoted as [r,0]) in a plane by specifying a distance r and an angle 0 from the origin of the system. There are several relationships between polar and rectangular coordinates, diagrammed in Figure 1-30. From the Pythagorean Theorem... 

To convert rectangular coordinates to polar coordinates, given the point (x,y), using the Pythagorean Theorem and the preceding equations. 

STRATEGY We calculate the density of the metal by assuming first that its structure is ccp (fee) and then that it is bcc. The structure with the density closer to the experimental value is more likely to be the actual structure. The mass of a unit cell is the sum of the masses of the atoms that it contains. The mass of each atom is equal to the molar mass of the element divided by Avogadro s constant. The volume of a cubic unit cell is the cube of the length of one of its sides. That length is obtained from the radius of the metal atom, the Pythagorean theorem, and the geometry of the cell. 

Figure 11-1 The distance between two points in a two-dimensional coordinate space is determined using the Pythagorean theorem.

And by using the Pythagorean theorem we can calculate the length of the hypotenuse ... 

For a face-centered cubic unit cell, the diagonal of a face is the hypotenuse of a right triangle and equals 4r where r is the radius of an Ir atom. From the Pythagorean Theorem,... 

Johannes Kepler as one of the two great treasures of geometry. (The other is the Pythagorean theorem.)... 

In right triangles, there is a special relationship between the hypotenuse and the legs of the triangle. This relationship is always true and it is known as the Pythagorean theorem. 

The Pythagorean theorem states that in all right triangles, the sum of the squares of the two legs is equal to the square of the hypotenuse leg2 + leg2 = hypotenuse2. 

The Pythagorean theorem is used to make measurements with right triangles. 

First make all units consistent. Change 4 feet to 48 inches. Now, use the Pythagorean theorem ... 

The distances that Shirley and Shelly walked are the a and b of a right triangle. The distance that they re apart, 17 miles, is the c value. Substituting into the Pythagorean theorem, the equation a2 + b2 = c2 becomes (y)2 + (2y - l)2 =172. 

In the case of Shirley and Shelly who walked away from each other in directions that are right angles from one another, you find that Shirley walked 8 miles and Shelly walked 15 miles. They re only 17 miles apart — even though they walked a total of 23 miles — because their journeys and the distance between them formed a right triangle. They didn t walk in opposite directions. As far as the accuracy of the solution, check out the Pythagorean theorem with the distances ... 

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