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Unformatted text preview: 72 CHAPTER 1. COORDINATES AND VECTORS A famous formula for the area of a triangle in terms of the lengths of its
sides is the second of two area formulas proved by Heron of Alexandria (ca.
75 AD) in his Metrica :
A= s(s − a)(s − b)(s − c) (1.25) where a, b and c are the lengths of the sides of the triangle, and s is the
semiperimeter
1
s = (a + b + c).
2
Equation (1.27) is known as Heron’s formula, although it now seems
clear from Arabic commentaries that it was already known to Archimedes
of Syracuse (ca. 287212 BC). In Exercise 15 and Exercise 16 we will
consider both of the area formulas given in the Metrica ; also, in Exercise 5
we will derive a vector formula for the area of a triangle based on the
discussion of the distance from a point to a line on p. 52.
Here, however, we will concentrate on ﬁnding a formula for the area of a
triangle in R2 in terms of the coordinates of its vertices. Suppose the
vertices are A(a1 , a2 ), B (b1 , b2 ), and C (c1 , c2 ). Using the side AB as the
base, we have
−
−
→
b = AB
and, letting θ be the angle at vertex A,
−
→
h = AC sin θ,
so 1−
−−
→→
AB AC sin θ.
2
To express this in terms of the coordinates of the vertices, note that
A (△ABC ) = −
−
→
→
→
AB = xAB − + yAB −
ı where
xAB = b1 − a1 yAB = b2 − a2 and similarly
−
→
→
→
AC = xAC − + yAC − .
ı ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Vectors, Formulas

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