508 triangle inequality

Mathematicians have proved that the number of differ-

ent integer triangles one can make with nmatchsticks is

given by the formula if nis even, and

if nis odd, where the brackets are used to indicate

rounding to the nearest integer.

The integer triangle 5-12-13 (a right triangle) has

the property that the numerical value of its area is the

same as its perimeter: A= 30, P= 30. The 6-8-10 trian-

gle is the only other right integer triangle with this

property. If one relaxes the condition of being right,

then there exist many nonright triangles with this prop-

erty. For instance, a 7-15-20 triangle has area and

perimeter each with the numerical value 42. (Use

Heron’s formula to compute its area.)

One can alternatively search for a pair of integer

triangles sharing the same value for perimeter and

same value for area. Again there are many such pairs.

For example, the 14-18-29 and 8-25-28 triangles each

have perimeter 61 and area 210√

–

22. The 45-94-94

and 49-84-100 also share the same perimeter and the

same area.

See also

AAA

/

AAS

/

ASA

/

SAS

/

SSS

;

BASE OF A POLYGON

/

POLYHEDRON

; B

ERTRAND

’

S PARADOX

;

CIRCUMCIRCLE

;

EQUILATERAL

;

FIGURATE NUMBERS

;

HYPOTENUSE

;

PEDAL

TRIANGLE

;

TRIANGULAR NUMBERS

.

triangle inequality The proposition that the sum of

the lengths of any two sides of a

TRIANGLE

is greater

than the length of the third side is called the triangular

inequality. Thus, if a, b, and care the three side-lengths

of a triangle, then each of the following relations hold:

a< b+ c

b< a+ c

c< a+ b

This result follows as a consequence of P

YTHAGORAS

’

S

THEOREM

. The converse proposition is also true: If

three numbers a, b, and csatisfy the three relations

above, then it is possible to draw a triangle with side-

lengths a, b, and c. (To see this, draw a line segment of

length a, draw a circle of radius bwith center one end-

point of the line segment, and a circle of radius cwith

the second endpoint as center. The three relations

ensure that these circles intersect at a point P. Then Pis

the apex of an a-b-c triangle with side-length aas the

base.) Thus the construction of a 7-9-12 triangle, for

example, is possible, but the construction of a 16-23-

40 triangle is not. (The number 40 is greater than 16 +

23.)

If any of the relations above is replaced by an equal-

ity, a= b+ c, for instance, then the corresponding trian-

gle is degenerate, meaning that its three vertices lie in a

straight line. This observation can be used as follows:

If the distance: from Adelaide to Darwin is

3,200 km, from Adelaide to Brisbane is 1,200

km, from Brisbane to Canberra is 600 km, and

from Canberra to Darwin is 1,400 km, then

one can only deduce that all four cities lie on

the same straight line.

The triangular inequality can be rephrased as follows:

The length of any one side of a triangle is less

than half the perimeter of the triangle.

(Adding ato both sides of the first inequality, for

instance, gives 2a< a+ b+ c. The right quantity is the

perimeter of the triangle.) This characterization allows

one to quickly identify all triangles with integer sides

having a given perimeter (as made with matchsticks,

for instance). For example, with 11 matchsticks one

can make four triangles given by the triples 5-5-1, 5-4-

2, 5-3-3, and 4-4-3. (Each number is less than half of

11.) Surprisingly, the count goes down if one adds

another matchstick to the collection—there are only

three integer triangles with perimeter 12: 5-5-2, 5-4-3,

and 4-4-4. Mathematicians have shown that the num-

ber of triangles one can produce with nmatchsticks is

if nis even and if nis odd, where the

angled brackets indicate rounding to the nearest integer.

triangular numbers See

FIGURATE NUMBERS

.

trigonometry Contrary to its name, the theory of

trigonometry is best motivated as a theory about

CIR

-

CLE

s, not triangles. (This, in fact, matches the historical

development of the subject.) Beginning with the sim-

plest circle imaginable, namely, a circle of radius 1 cen-

tered about the origin, one defines two functions, sine

and cosine, simply as the x- and y-coordinates of a