Download Chapter 5 - TeacherWeb

Chapter 12: Ratios , Proportion and Similarity
Definitions:
If the 2 means of a proportion are equal, the mean
proportional is one of the means. (also called the geometric
mean)
Two line segments are divided proportionately when the ratio of
the lengths of the parts of one segment is equal to the lengths
of the parts of the other.
Two Polygons are Similar if there is a one-to-one
correspondence between vertices such that:
 All pairs of corresponding angles are congruent
 The ratios of the lengths of all pairs of corresponding
sides are equal.
If two polygons are similar, then their corresponding angles are
congruent and their corresponding sides are in proportion.
If two polygons have corresponding angles that are congruent
and corresponding sides that are in proportion, then the
polygons are similar.
If two polygons have corresponding angles that are congruent
and corresponding sides that are in proportion, then the
polygons are similar.
The ratio of any length in the transformed figure to the
corresponding length in the original figure is called the scale
factor (k) or constant of dilation.
The ratio of the corresponding sides of similar figures is called
the ratio of similitude.
The point of concurrence of the medians of a triangle is called
the Centroid
Pythagorean Triples
 3,4, 5
 45-45-90 (1:1: 2 )
 30-60-90 (1: 3 : 2)
Postulates, Theorems and Corollaries
Theorem
Corollary
Corollary
Corollary
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
Corollary
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
Theorem
In a proportion, the product of the means is equal to the product of
the extremes.
In a proportion, the means may be interchanged.
In a proportion, the extremes may be interchanged.
If the products of 2 pairs of factors are equal, the factors of one pair
can be the means, and the factors of the other can be the extremes
of a proportion.
A line segment joining the midpoints of two sides of a triangle is
parallel to the third side and its length is one-half the length of the
third side.
If two line segments are divided proportionately, then the ratio of the
length of a part of one segment to the length of the whole is equal to
the ratio of the corresponding lengths of the other segment.
If the ratio of the length of a part of one line segment to the length of
the whole is equal to the ratio of the corresponding lengths of
another line segment, then the two segments are divided
proportionately.
Two triangles are similar if the ratios of corresponding sides are
equal. (SSS~)
Two triangles are similar if the ratios of two corresponding sides are
equal and the corresponding angles included between these sides
are congruent. (SAS~)
Two triangles are similar if three angles of one triangle are congruent
to three corresponding angles of the other.
Two triangles are similar if two angles of one triangle are congruent
to three corresponding angles of the other.
If two triangles are similar the lengths of corresponding altitudes
have the same ratio of any two corresponding sides.
If two triangles are similar the lengths of corresponding medians
have the same ratio of any two corresponding sides.
If two triangles are similar the lengths of corresponding angle
bisectors have the same ratio of any two corresponding sides.
Any two medians of a triangle intersect in a point that divides each
median in the ratio of 2 : 1.
The altitude to the hypotenuse of a right triangle divides the triangle
into two triangles that are similar to each other and to the original
triangle.
The altitude is the mean proportional between the measures of the
segments of the hypotenuse.
The length of each leg of a right triangle is the mean proportional
between the hypotenuse and the segment of the hypotenuse that is
adjacent to that leg (projection of the leg).
Pythagorean If a triangle is a right triangle, then the square of the length of the
Theorem
longest side is equal to the sum of the squares of the lengths of the
other two sides (legs).
Converse of If the square of one side a triangle is equal to the sum of the squares
the
of the lengths of the two other sides, then the triangle is a right
Pythagorean triangle.
Theorem