Hero

1
Heros and Brahmaguptas FormulasLesson 11.8
2
Hero of Alexandria

• He was an ancient Greek mathematician and
  engineer who was born in 10 AD.
• He invented many different machines.
• Steam powered device called an aeolipile.
• A vending machine. (When the coin went in, holy
  water came out.)
• A windwheel operating organ. (This is the first
  instance of using wind power in history.)
• And many other machines
• His contribution to mathematics was with the
  imaginary number, a method to compute square
  roots iteratively and Herons Formula for find
  the area of any triangle.

3
a
b
Heros Formula Used to calculate the area of
any triangle.
c
Theorem 111 A?
Where a, b, c are lengths of the sides and s
semi-perimeter
S a b c
2
4
4
Find the area of a triangle with sides 3, 6, and
7. First find the simiperimeter S 3 6
7 2 S 8
5
Substitute into Heros Formula.
A
6

• Brahmagupta lived between 598 and 668 AD in
  India.
• He was an Indian mathematician and astronomer.
• He was first to use zero as a number.
• He was first to say that two negative numbers
  multiplied together equal a positive number.
• He gave the solution to the general linear
  equation.
• He gave two solutions to the general quadratic
  equation.
• He finds Pythagorean triples.
• He created the Brahmagupta Formula.
• He created or discovered many other things in
  math.

7
Brahmaguptas Formula is a way to find the area
of an inscribed quadrilateral.
These are known as cyclic quadrilaterals.
Theorem 112 Acyclic quad
Where a, b, c, d are sides of the quadrilateral,
and s the semiperimeter.
8
Brahmaguptas Formula
Find the area of the inscribed quadrilateral.
Show all steps. First find the semi-perimeter. S
7 6 2 9 2 S
12 Substitute into Brahmaguptas Formula solve.
6
7
2
9
9
A