Proportions

1
Proportions Similarity

• Objectives to use
• -Ratios proportions
• -Similar figures
• -Proportional parts of triangles

2
Using Proportions

• a d are the extremes of the proportion
• b c are the means of the proportion
• The product of the meansthe product of the
  extremes

• a/b is a ratio of two numbers
• a b c is a ratio of 3 numbers.
• A ? that has a 345 ratio of the sides and a
  perimeter of 24? 3x 4x 5x 24 sets up the
  solution.

3
Similar Polygons

• Two polygons are similar iff corr corr
  sides proportional
• Sides are proportional if their ratios are all
  equal
• eg

5
3
12.5
7.5
4
10

• The ratio is called the scale factor
• Enlarging or shrinking the polygon is called
  dilation
• If scale factor gt1?it enlarges, if sf lt1 ?it
  shrinks

4
Similar Triangles

• AA If
• SSS If the measures of the corr sides of 2 ?s
  are proportional
• SAS If 2 sides of a ? are proportional to 2 corr
  sides of a 2nd ? the included angle is

THEN the 2 ?s are similar

• Similarity is reflexive, symmetric, and
  transitive.

5
Parallel lines proportional parts
C
B
gt
D
gt
A
E

• A line // to a side of a ? that intersects the 2
  other sides in 2 distinct points (B D) iff it
  separates these sides into proportional lengths
• A line joining the midpoints of 2 sides of a ? is
  // to the 3rd side its length ½ the 3rd side
• 3 or more // lines intersect 2 transversals, they
  cut them proportionately

6
Parts of similar triangles

• The perimeters
• corr. altitudes
• corr. angle bisectors
• corr. medians
• are each proportional to corresponding sides
• Angle bisectors create the following proportions

C
A
B
D