Shape 

Copyright David & Cynthia Thomas, 2009

Chapter Two: Summary

Chapter Two began by asking four questions.  Those questions are …

1. What is the relationship between area and perimeter in triangles? 

2. What information is needed to determine the area of a triangle? 

3. What relationships exist between the sides, angles, and interior lines and points of a given triangle?

4. What relationships exist between triangles with different side and angle characteristics?

The answers to those questions may now be summarized as follows:

1. The perimeter of a triangle is a property of the one-dimensional loop created by the segments comprising the triangle.  Since segment lengths are measured in one-dimensional units (e.g., centimeters, inches, and so on), the perimeter is also expressed in one-dimensional units.  The area of a triangle is a property of the interior of the triangle, a two-dimensional space.  As such, area is measured in terms of square units (e.g., cm², in², and so on).  For any given perimeter, the triangle with maximum area is always equilateral.  All other triangles with identical perimeters will have areas greater than zero but less than that of the equilateral triangle. 

2. Among many possible methods, the area of a triangle may be computed using …

· Additive or subtractive strategies in the case of lattice triangles;

· The formula A = ½bh where b is any side of the triangle and h is the altitude to that side from the opposite vertex.

3. Many relationships exist between the sides, angles, and interior points and lines of a triangle, including …

• The Pythagorean Theorem a² + b² = c².
• The perpendicular bisectors of the sides of a triangle intersect in a common point called the circumcenter which is equidistant from the vertices.
• The angle bisectors of a triangle intersect in a common point called the incenter which is equidistant from the sides.
• The medians of a triangle intersect in a common point called the centroid, also known as the center of gravity.

4. Triangles may be sorted by their angle characteristics (acute, right, and obtuse) and their side characteristics (equilateral, isosceles, and scalene).  All triangles belong to one of seven categories: Equilateral, isosceles acute, isosceles right, isosceles obtuse, scalene acute, scalene right, and scalene obtuse.