Chapter 8 Summary
Misc
Definitions
- ratio
- a quotient of two numbers, e.g. 2/3
- proportion
- an equation stating that two or more ratios are equal, e.g. 2/3 = 4/6 = 20/30
- means
- The 2nd and 3rd terms of a proportion
- extremes
- The 1st and 4th terms of a proportion
- mean proportion
- A proportion in which the means are equal: a/b = b/c.
- geometric mean or mean proportional
- A mean in a mean proportion
- similar figures
- Figures with the same shape but not necessarily the same size.
- similar polygons
- Polygons for which (a) the ratios of measures of corresponding sides are equal, and (b) the measures of corresponding angles are equal.
- dilation
- Enlarging a figure to get a larger similar figure.
- reduction
- Shrinking a figure to get a smaller similar figure.
Theorems
- Th. 59
- In a proportion, the product of the means equals the product of the extremes. This is just a way of saying that cross-multiplying works: if a/b = c/d then ad = bc.
- Th. 60
- Given nonzero numbers a, b, c and d, if ad = bc, then a/b = c/d, b/a = d/c, a/c = b/d, and c/a = d/b. This is how we know we can turn a proportion "on its side": if a/b = c/d then a/c = b/d.
- Th. 61
- The ratio of the perimeters of similar polygons equals the ratio of any pairs of corresponding sides.
- Th. 62 (AA)
- If there is a correspondence between the vertices of two triangles such that two corresponding angle measures are equal, then the triangles are similar.
- Th. 63 (SSS~)
- If there is a correspondence between the vertices of two triangles such that the ratios of corresponding sides are equal, then the triangles are similar.
- Th. 64 (SAS~)
- If there is a correspondence between the vertices of two triangles such that the ratios of two corresponding sides are equal and the included angles are congruent, then the triangles are similar.
- Th. 65 (Side-splitter)
- If a line is parallel to a side of a triangle and intersects the other two sides, it divides those sides proportionally.
- Th. 66
- If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.
- Th. 67 (Angle Bisector)
- The bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides.