(10 – x)(10 + x) = 100 – x2. We see now that the area
is maximal precisely when xis zero, that is, the figure
is a square.
An exercise in geometry shows that of all triangles of
a given perimeter, the equilateral triangle has largest area.
See also
SOAP BUBBLES
; S
TEINER POINT
.
isosceles trapezoid (isosceles trapezium) A trapezoid
with the two nonparallel sides of equal length is called
an isosceles trapezoid. The figure has a line of symmetry
perpendicular to the parallel sides. Any isosceles trape-
zoid can be considered a truncated
ISOSCELES TRIANGLE
.
See also
TRAPEZOID
/
TRAPEZIUM
.
isosceles triangle A triangle possessing two sides of
equal length is called isosceles. The name is derived
from the Greek terms iso meaning “same” and skelos
meaning “leg.” Some scholars insist that a triangle be
called isosceles only if two, but not all three, of its sides
are equal in length, but this is really only a matter of
taste. (A triangle with all three sides equal in length is
called equilateral.) The base angles of an isosceles tri-
angle are equal. This can be established as follows:
Suppose triangle ABC is isosceles with sides AB
and BC equal in length. Regard a diagram of
this triangle as the picture of two triangles,
ABC and CBA superimposed on top of one
another. These two triangles share pairs of sides
that match in length, and share a common
angle (at B) between those two sides. Thus, by
the SAS principle from the study of similarity,
the two triangles are congruent. Consequently,
all corresponding angles between the triangles
match. In particular, the angle at vertex Amust
match the angle at vertex C.
If one slides the vertex Bin a direction parallel to the
base AC, then the perimeter of the triangle changes
(although the
AREA
of the figure, A= ×base ×height
does not). The study of
OPTIMIZATION
shows that the
sum of distances |AB| + |BC| is at a minimum when the
triangle is isosceles. We have:
Of all triangles with a given base and a fixed
height, the isosceles triangle has the least
perimeter.
To minimize the entire perimeter of a triangle without
changing its area, one can adjust each vertex of the
triangle (by sliding the vertex along a line parallel to
the opposite side) so that it is an apex of an isosceles
triangle. This produces an equilateral triangle and
thereby establishes:
Of all triangles of a fixed area, the equilateral
triangle has the least perimeter.
It is possible to arrange five points in a plane so
that any three points chosen at random among them
form the vertices of an isosceles triangle. (The points lie
at the vertices of a regular pentagon.) It is impossible to
accomplish the same feat with six or more points.
See also
AAA
/
AAS
/
ASA
/
SAS
/
SSS
;
CONGRUENT FIGURES
;
TRIANGLE
.
iteration The repeated application of a mathematical
procedure in which each step is applied to the output
of the previous step is called iteration. For example,
H
ERON
’
S METHOD
for computing the square roots of
numbers is an iterative procedure.
The iterates of a function f, beginning with the
value x0, are defined by the sequence of values:
x1= f(x0), x2= f(x1) = f(f(x0)), x3= f(x2) = f(f(f(x0))),…
If this sequence converges to a value a, say, then:
a= limn→∞ xn= limn→∞ f(xn–1) = f(a)
(We are assuming here that fis a
CONTINUOUS FUNC
-
TION
.) Thus iteration provides a means for solving
1
–
2
iteration 285
Isosceles triangle