Greenwood press

86
PROPORTIONS

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PYTHAGOREAN THEOREM
87
PYTHAGOREAN THEOREM
The Pythagorean theorem states that the sum of the squares of the legs of a
right triangle, 
a
2
+ b
2
, is the same as the square of its hypotenuse, 
c
2
. There are
over 100 proofs of the Pythagorean theorem, many of which show that the sum
of the areas of squares on the legs is equal to the area of the square on the hypot-
enuse, as shown in the figure below. Conversely, any triangle that has sides that
are related by the equation 
a
2
+ b
2
= c
2
must have a right angle opposite the
longest side.
The Pythagorean theorem is useful on a baseball diamond for several rea-
sons. Since the bases are each 90 feet apart in the form of a square, the theorem
helps us find the distance the catcher has to throw the ball to second base when
a runner is trying to steal. The right triangle formed would be with half of the
infield, where the legs of the triangle are the base paths of 90 feet each, and the
hypotenuse is from home plate to second base. The hypotenuse can be found by
solving the equation 
90
2
+ 90
2
= c
2
. Solving for 
c will show that the throw is
about 127.3 feet. This information is useful, because it will give coaches an idea
about how hard the catcher needs to be able to throw a ball accurately in order to
throw a runner out. If the catcher throws a ball at about 70 miles per hour, then
it will only take about one-and-a-quarter seconds for the ball to reach the base. 
The geometry of rhombuses and the Pythagorean theorem can be used to
show that the center of the pitcher’s mound is not in the pathway of the ball when
it is thrown from third to first base. The diagonals of the square running-path
between the bases are perpendicular bisectors of each other, forming congruent
right triangles in the center. If the pitcher was placed at the intersection of the
diagonals, he might get hit by a throw from the third baseman. To avoid contact,
the pitcher needs to be placed closer to home plate than this intersection. The
Pythagorean theorem gave the distance between home and second base to be
127.3 feet. The pitcher must be closer to home plate than 63.6 feet. The actual
placement of the center of the pitching mound is 60.5 feet from home plate.
The Pythagorean theorem is used to approximate the distance of two nearby
towns on a map. Changes in the earth’s curvature are minimal within short
ranges, so the latitude and longitude positions can serve as points on a coordinate
plane. For example, suppose that Smithsville is five miles north and two miles
east of Laxtown. The two cities would be 5.39 miles away on a map, represent-
ing the distance that the “crow flies.” This distance can be determined by solv-
ing the equation 
5
2
+ 2
2
= d
2
that is determined with the Pythagorean theorem.
The Pythagorean theorem illustrates
that the sum of the areas of the
squares connected to the legs of a
right triangle is equal to the area of
the square connected to the
hypotenuse of a right triangle.

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Carpenters use Pythagorean triples to verify that they have right angles in
their work. For example, a carpenter making a cabinet can perfectly align pieces
of wood in a right angle with the use of only a tape measure. Using the Pytha-
gorean triple {3,4,5}, or any multiple such as {12,16,20}, the carpenter can place
a mark on the bottom after 12 inches, a mark on the side after 16 inches, and
rotate the intersecting boards at its hinge until the distance between the markings
is 20 inches. A triangle with sides of 12, 16, and 20 inches is a right triangle,
since
12
2
+ 16
2
= 20
2
.
Construction workers building along the sides of mountains use the Pythagor-
ean theorem to determine the amount of supplies needed to create a railroad track
for a funicular or a cable line for a gondola. The horizontal and vertical distances
from the foot of the mountain to its top can be determined on a map, forming the
legs of a right triangle that can be drawn in the mountain’s center. The third side of
the triangle, the hypotenuse, represents the walk up the mountain, which never has
to be physically measured, since it can be found using the Pythagorean theorem.
The visible distance to a horizon can be found with the Pythagorean theorem,
given that the radius of the earth is 6,380 km. Inside a 100-meter-tall lighthouse,
a night watchman or the coast guard may be interested in the distance a ship is
from shore when seen at the horizon. This information can be readily found, since
the horizon distance is perpendicular to the radius of the earth, forming a right tri-
angle into the center of it, as shown below. The viewing distance inside the top of
the lighthouse is then the solution to the equation 
6380000
2
+ b
2
= 6380100
2
, a
value of over 35 km!
Extensions of the Pythagorean theorem provide distances in three or more
dimensions. If a rectangular box has dimensions of length 
L, width W , and
height
H, the main diagonal has a length given by d
2
= L
2
+ W
2
+ H
2
. Can a
42-inch-long umbrella be packed into a carton that is 40 inches long, 10 inches
wide, and 10 inches high? According to the three-dimensional Pythagorean the-
orem, the diagonal is about 42.43 inches long. Yes, it should just barely fit. (See