A mathematical
theory is built from a system of definitions, axioms (assumed
truths), and theorems. For Euclidean geometry, area is such a
theory built from definitions, axioms, and theorems.
Area
Axioms
For every polygon P, a nonnegative real
number denoted area(P) can be associated with the
polygon such that:
Axiom
1: Congruent polygons have equal areas.
Axiom 2:
If P can be cut into two non-overlapping
polygons P1 and P2, then
area(P) = area(P1) +
area(P2).
Axiom 3: If
p is a square with sides of length 1, then area(P)
= 1.
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Area
Theorems
From the above axioms, these theorems can be
proven:
Theorem
1: Let ABCD and EFGH be
rectangles such that 
and BC = a * FG, for some positive real number
a. Then area (ABCD) = a * area (EFGH).
Corollary 1:
If ABCD is a rectangle with AB = 1 and BC = h, then
area (ABCD) = h.
Corollary 2:
If ABCD is a rectangle with AB = b and BC = h, then
area (ABCD) = bh.
Theorem
2: The area of a triangle is 1/2
* base * height.
Theorem
3: Let  be
a right triangle with right angle at C, Denote AB =
c, BC = A and AC = b. Then,
a2 +b2 =
c2
Corollary 3:
Let be
a right triangle and denote the lengths of the
sides
AB = c, BC = A and AC = b. If a2
+b2 = c2, then <C =
900.
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Sketchpad's Area
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11
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The Problem:
Given any triangle, say the
on the right.
Show that area(ADF) = area(FBE).
Can you prove this?
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Steps to find
Area:
1) Construct any triangle
2) Construct the midpoints of two sides
and construct the medians.
3) Select the three points that form
one of the little triangles (by holding
the shift key as you select the
points).
4) With the three points selected,
choose
Construct -> Polygon
Interior.
5) The interior of the polygon will
shimmer to denote that it has been
selected. While it is selected, choose
Measure -> Area.
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NOTE1: To change the color of the
polygon, use Color and Shade
under the Display menu.
NOTE2: To select a polygon, simply
click on it once and it will begin to
shimmer.
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