DILATIONS
Reflections, translations, glide reflections, and rotations are
examples of
"congruence mappings."
Dilations are examples of "similarity mappings:" expansions or contractions
of the size of objects according to some scale factor.
DILATIONS
When the scale factor is greater than 1, the dilation is an "expansion."
When the scale factor is less that 1, the dilation is a "contraction."
RATIO AND PROPORTION
RATIO: a comparison of numbers by division.
ratios are usually expressed
in lowest terms
PROPORTION: two ratios that are equal to each other.
PROPERTIES OF PROPORTIONS
The product of the extremes equals the product of the means.
PROPERTIES OF PROPORTIONS
You can add or subtract the number "1" from both sides of a
proportion to get an equivalent proportion.
SIMILAR POLYGONS
SIMILAR POLYGONS: polygons whose corresponding angles are congruent
and
whose corresponding sides are proportional
SIMILAR POLYGONS : SCALE FACTOR
If two polygons are similar, the "scale factor" is the
ratio of the lengths of
any two corresponding sides.
SIMILAR TRIANGLES: AA SIMILARITY POSTULATE
POSTULATE : AA SIMILARITY POSTULATE: If two angles of one triangle
are
congruent to two angles of another triangle, then the two triangles are similar.
(remember: "similar" means "the sides are proportional.")
SIMILAR TRIANGLE THEOREMS
THEOREM : SAS SIMILARITY THEOREM. If two sides of two triangles
are in proportion and the included angles are congruent, then the two triangles
are
similar.
SIMILAR TRIANGLE THEOREMS
THEOREM : SSS SIMILARITY THEOREM: If the sides of two triangles
are in proportion, then the two triangles are similar.
PROPORTIONAL LENGTHS
AC and QR are divided proportionally if:
PROPORTIONAL LENGTHS
THEOREM : TRIANGLE PROPORTIONALITY THEOREM: If a line parallel
to one side of a triangle intersects the other two sides, then it divides
those sides proportionally.
PROPORTIONAL LENGTHS
Because of the TRIANGLE PROPORTIONALITY THEOREM, any proportion
equivalent to : is also true.
PROPORTIONAL LENGTHS
COROLLARY: If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
PROPORTIONAL LENGTHS
THEOREM : TRIANGLE ANGLE-BISECTOR THEOREM: the bisector of the
angle of a triangle divides the opposite side into segments that are proportional
to the other sides of the triangle.
SIMILARITY IN RIGHT TRIANGLES
DEFINITION: GEOMETRIC MEAN:
SIMILARITY IN RIGHT TRIANGLES
THEOREM : The altitude to the hypotenuse of a right triangle forms
two triangles that are similar to each other and to the original trianle.
SIMILARITY IN RIGHT TRIANGLES
COROLLARIES: the following ratios remain constant when comparing
any of the three similar triangles:
short leg short leg long leg
long leg hypontenuse hypontenuse
SIMILARITY IN RIGHT TRIANGLES
COROLLARY: The altitude to the hypotenuse of a right triangle is
the geometric mean between the segments of the hypotenuse.
SIMILARITY IN RIGHT TRIANGLES
COROLLARY : The altitude to the hypotenuse of a right triangle
also sets up the following proportion.
RATIOS OF AREAS
THEOREM :
If the scale factor of two similar polygons is a : b, then
1) the ratio of their perimeters is a : b
2) the ratio of the areas is a^2 : b^2
AREAS AND VOLUMES OF SIMILAR SOLIDS
USE THIS THEOREM TO COMPARE TWO SIMILAR SOLIDS:
THEOREM : if the scale factor of two similar solids is a : b, then:
(1) the ratio of corresponding perimeters is a : b
(2) the ratio of base areas, lateral areas, and
total areas is a^2 : b^2
(3) the ratio of volumes is a^3 : b^3