Proportion in mathematics is a fundamental concept that relates to the consistent ratio between two quantities. It is essential in various fields, including physics, where Ohm's law illustrates direct proportionality with voltage and current. Inverse proportionality is also discussed, where one variable increases as the other decreases, represented by a hyperbolic curve on a graph. The concept extends to geometry, where it defines the similarity between figures and is described by scale factors.
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Proportion establishes a consistent ratio between two changing quantities
Symbolization
Proportions are symbolized by the sign \(\propto\)
Example
Ohm's law in physics, which states that voltage is directly proportional to current, is denoted as \(V \propto I\)
Proportions can be direct or inverse, depending on the relationship between the two quantities
Direct proportionality occurs when two variables change in tandem, maintaining a constant ratio
Direct proportionality is represented by a straight line on a graph that intersects the origin
The weight of a string is directly proportional to its length, with the constant of proportionality representing the weight per unit length
Inverse proportionality occurs when one variable increases as the other decreases
Inverse proportionality is represented by a hyperbolic curve on a graph
The relationship between the weight and length of a string is inverse, with the constant of proportionality representing the weight per unit length
Proportionality in geometry defines the similarity between figures based on corresponding angles and proportional dimensions
Length Scale Factor
The length scale factor relates corresponding lengths in similar figures
Area Scale Factor
The area scale factor is the square of the length scale factor
Volume Scale Factor
The volume scale factor is the cube of the length scale factor
Similar geometric figures have corresponding angles that are congruent and sides that are in proportion, with the scale factor representing the ratio of corresponding lengths
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Inverse Proportionality Definition
Relationship where one variable increases as the other decreases.
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Mathematical Expression of Inverse Proportion
c is inversely proportional to d can be written as c ∝ 1/d or c = k/d.
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Graphical Representation of Inverse Proportion
A hyperbolic curve that approaches but never touches the axes.
