Chapter 5: Similar Triangles 5.1: ratio, Rate, Proportions 5.2: Similar

Chapter 5: Similar Triangles
5.1: ratio, Rate, Proportions
5.2: Similar Polygons
5.3: Proving Triangles Similar
5.4: The Pythagorean Theorem
5.5: Special Right Triangles
5.6: Segments divided Proportionally.
Ratio
π‘Ž
Is a quotient 𝑏 .
Rate:
Is a quotient that relates two quantities that ar incommensurale
Commensurable quantities
Is
Incommensurable quantities
Is
Proportion:
Is a statement that equates two ratios or two rates
Property 1
π‘Ž
𝑐
In a proportion, the product of the means equals the product of the extremes; that is, if 𝑏 = 𝑑, then π‘Ž βˆ™ 𝑑 = 𝑏 βˆ™ 𝑐.
Definition:
π‘Ž
𝑐
In a proportion, the product of the means equals the product of the extremes; that is, if = , then π‘Ž βˆ™ 𝑑 = 𝑏 βˆ™ 𝑐.
𝑏
𝑑
Definition
The nonzero number b is the geometric mean of a and c if
π‘Ž
𝑏
𝑏
= 𝑐 , or
𝑐
𝑏
𝑏
=𝑐.
Property of Extended Ratio:
Unknown quantities in the ratio π‘Ž: 𝑏 = 𝑐: 𝑑 can be represented by π‘Žπ‘₯, 𝑏π‘₯, 𝑐π‘₯, and 𝑑π‘₯.
Property 2 – Alternative Forms of Proportions
In a proportion, the means and the extremes (or both) may be interchanged; that is, if
Property 3 – Sum and Difference Properties of a Proportion
π‘Ž
𝑐
π‘Ž+𝑏
𝑐+𝑑
π‘Žβˆ’π‘
π‘βˆ’π‘‘
If 𝑏 = 𝑑 ,(when 𝑏 β‰  0 and 𝑑 β‰  0), then 𝑏 = 𝑑 , and 𝑏 = 𝑑 .
π‘Ž
𝑐
𝑏 𝑑
𝑐
= 𝑑 , 𝑏 = π‘Ž , or
𝑑
𝑐
𝑏
=π‘Ž.