Shape 

Copyright David & Cynthia Thomas, 2009

 

Chapter Five: Summary

 

Chapter Five began by asking three questions.  The answers to those questions may be summarized as follows: In coordinate geometry,

·        Questions #1:

o      Segment length: If a segment has endpoints A(a1, a2) and B(b1, b2), the length of the segment is given by the formula .

o      Segment midpoint: If a segment has endpoints A(a1, a2) and B(b1, b2), the midpoint of the segment is given by the formula .

o      Slope of a line or segment: If a segment has endpoints A(a1, a2) and B(b1, b2), the slope of the segment is given by the formula .

o      Parallel lines or segments:  Lines and/or segments that are parallel have the same slope. 

o      Perpendicular lines or segments: Lines and/or segments that are perpendicular have slopes that are negative reciprocals of one another (i.e., the product of their slopes equals -1).

·        Questions #2:

o      Triangle area: If the coordinates of the vertices of a triangle are represented as , the area of the triangle is given by .

o      Polygons area: Since every n-gon may be divided into n-2 triangles, the triangle area formula is used n-2 times to compute these areas and their sum computed.

·        Question #3:

o      Translations are represented using slide arrows.  The differences in the x- and y-coordinates of the endpoints of the slide arrow are added to the x- and y-coordinates, respectively, of the polygon vertices to obtain the coordinates of the image of the polygon.

o      Reflections are represented reflecting lines, or mirrors.  Image points are located the same distance from the mirror as the object points.