Chapter 6 Proportions and Similarity

Section 6.1 Proportions

Ratio--a comparison of 2 quantities  example:  3:4 or 3/4

Example:

    If a class of 20 has 14 boys in it...

    write the ratio of boys to students

    write the ratio of girls to students

    write the ratio of boys to girls

 

 

 

Simplifying Ratios: (write both quantities in the same units before you simplify the fraction)

There are conversions on p730 and p731 into your textbook and on page r10 in your student planner if you need to look any of them up.

    Examples:

    3 pounds : 12 ounces

 

    6 days : 2 weeks

 

    20 feet : 3 years

 

    1 mile : 2640 feet

 

 

 

    Extended Ratio--comparison of 3 or more numbers 
           example:    3:4:5

 

Example:

    What is the ratio of the angles in the triangle below?

   

 

 

Example:

Three sides of a triangle are in the extended ratio of 9:8:7 and the triangle has a perimeter of 144.

Find the length of each side and list them from shortest to longest.

 

 

 

 

Example:

Three angles of a triangle are in the extended ratio of 5:7:8

Find each angle measure and list them from smallest to largest.

 

 

 

 

Proportion--an equality relating two ratios  ex 3/4 = 6/8

Can be solved by cross multiplication

Examples:

 

                           

 

 

 

 

 

 

 

 

 

Means and Extremes--Names for the parts of a proportion.

example

 3  =  x
5      15

 3 and 15 are called the extremes of the proportion.
5 and x are called the means of the proportion.

Values, variables or expressions in the upper left and lower right positions of the proportion are the extremes.

Values, variables or expressions in the lower left and upper right positions of the proportion are the means.

 

 

 

 

The Geometric Mean is found using this concept.

Find the Geometric Mean of 4 and 9

 4  =  x
x     
9

 

 

Some Tricks with proportions:

1.  The means (or extremes) can be exchanged and the proportion is still true.

 

 

 

2.  Reciprocal Rule

 

 

 

3.  Add up (or subtract up) Rule

 

 

 

 

 

 

Examples:

A model house is 30 inches long and the actual house is 48 feet long.  Find the scale factor. (this is a ratio)

 

A school has 108 teachers and 1900 students.  What is the ratio of students to EACH teacher? (how many students per 1 teacher?)

 

 

Find the ratio of AE to DI:

Find the ratio of IF to CF:

Find the ratio of CK to DF:

 

 

Section 6.2 Similar Polygons  

Polygons of the same shape, but different in size are considered to be similar.  (example:  stop signs {octagons} of different sizes. ) 

 

 

 

Corresponding angles must all be congruent and
corresponding sides
must all be proportional.

 

 

 

Example

 

 

Example

 

 

Example

 

Are the two rectangles similar? If so, what is the scale factor? 

 

 

 

 

     

 

 

 

 

 

 

Section 6.3 Similar Triangles

Angle Angle (AA) Similarity:  If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Side Side Side (SSS) Similarity:  If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

Side Angle Side (SAS) Similarity: If the measures of two sides of one triangle are proportional to two corresponding sides of a second triangle and the included angles are congruent, then the triangles are similar.

Example

 A person that is 2 yards tall casts a shadow of 3 yards, while at the same time a tree casts a shadow of 40 yards.

What is the height of the tree?

 

            

 

Section 6.4 Parallel Lines and Proportional Parts

Triangle Proportionality Theorem:  If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides into segments of proportional lengths. 

 

 

 

 

 

 

 

 

 

 

 

 

 

Triangle Proportionality Theorem Converse:  If a line intersects two sides of a triangle and divides these sides into corresponding segments of proportional lengths, then the line is parallel to the third side of the triangle.

              

Is the red segment parallel to the base of the triangle?

 

 

Midsegment--a segment whose endpoints are the midpoints of two sides.

Triangle Midsegment TheoremA midsegment of a triangle is parallel to one side of a triangle and is 1/2 the length of that side.

 

 

 

 

 

 

Proportional Parts of Parallel Lines:  The transversals are cut proportionally by the parallel lines.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Section 6.5 Parts of Similar Triangles

 Proportional Perimeters Theorem:  If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides.

 

Corresponding Altitudes:  If two triangles are similar, then the measures of corresponding altitudes are proportional to the measures of corresponding sides.

 

Corresponding Angle Bisectors:  If two triangles are similar, then the measures of corresponding angle bisectors are proportional to the measures of corresponding sides.

 

Corresponding Medians:  If two triangles are similar, then the measures of corresponding medians are proportional to the measures of corresponding sides.

 

Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.