506 triangle

Turning a pencil inside a triangle

Establishing the converse of Pythagoras’s theorem

This demonstration shows that the net effect of

rotating a pencil through each of the three angles of a

triangle is to rotate the pencil through half a turn.

Thus the three angles of the triangle do indeed sum to

180°. Moreover, it is clear that this argument would

work for any triangle one could draw. A similar argu-

ment shows that the sum of angles in any quadrilat-

eral is 360° in any five-sided polygon, 540°, and, in

general, the sum of angles in any N-sided shape is

(N– 2) ×180°.

Triangles are classified according to the relative

lengths of their sides: A triangle is scalene if each side is

of a different length, isosceles if at least two sides are

equal in length, and equilateral if all three sides are the

same length. A study of the SAS principle shows that

the angles

OPPOSITE

the two sides equal in length in an

isosceles triangle have equal measure. As an equilateral

triangle is also isosceles, it follows that an equilateral

triangle is

EQUIANGULAR

, that is, all three angles have

the same measure (of 60°).

Triangles can alternatively be classified according to

their angles. An acute triangle has all three angles less

than 90°in measure, a right triangle contains precisely

one angle equal to 90°, and an obtuse triangle contains

one angle of measure greater than 90°. (It is not possi-

ble for a triangle to possess two obtuse angles.)

If a picture of a triangle is oriented so that one side

of the triangle is horizontal, then that side is called the

base of the triangle. The height of the triangle is the ver-

tical distance between the base of the figure and the ver-

tex of the triangle that does not lie on the base. A study

of

AREA

shows that, if oriented appropriately, the inte-

rior of a triangle occupies half the interior of the rectan-

gle that encloses the triangle, with the base as one side

of the rectangle. Thus the area of a triangle is given by

the formula:

If the three sides of the triangle are of lengths a, b, and

c, then H

ERON

’

S FORMULA

also shows that the area of

the triangle is given by:

where is the semiperimeter of the triangle.

P

YTHAGORAS

’

S THEOREM

shows that if a right tri-

angle has side-lengths a, b, and c, with the sides aand

bsurrounding the right angle, then a2+ b2= c2. The

TRIANGLE INEQUALITY

follows as a consequence.

For an arbitrary triangle with side-lengths a, b, and

c, if the angle between sides aand bis acute (that is, less

than 90°), then a2+ b2> c2. If, on the other hand, the

angle between sides aand bis obtuse (greater than 90°),

then a2+ b2< c2. This follows from the

LAW OF

COSINES

. It can also be seen algebraically as follows:

For the diagram below left, the angle between

sides aand bis acute. We clearly have a> xand

b> y, and so a2+ b2> x2+ y2= c2. For the dia-

gram below right, the angle between sides aand

bis obtuse. We clearly have x> a. Consequently:

c2= x2+ y2

= x2+ (b2– (x– a)2)

= b2+ 2ax – a2

< b2+ 2a· a– a2= b2+ a2

These observations provide a proof of the

CONVERSE

of

Pythagoras’s theorem:

If, for a triangle with side-lengths a, b, and c,

we have a2+ b2= c2, then the triangle is a right

triangle with right angle between the sides of

length aand b.