In the realm of mathematics, particularly in algebra and geometry, the Multiplication and Division Postulates are crucial principles. These postulates pertain to manipulating expressions, equations, and inequalities in order to isolate and solve for variables. Here are the definitions for the Multiplication \& Division Postulates:
Multiplication Postulate
If \( a = b \), then \( a \times c = b \times c \)
If equal quantities are multiplied by the same quantity, the products are equal.
If congruent segments are multiplied by the same quantity, the products are congruent.
If congruent angles are multiplied by the same quantity, the products are congruent.
Division Postulate
If \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \)
If equal quantities are divided by the same non-zero quantity, the quotients are equal.
If congruent segments are divided by the same non-zero quantity, the quotients are congruent.
If congruent angles are divided by the same non-zero quantity, the quotients are congruent.
Algebraic Context
The Multiplication and Division Postulates serve as key tools to solve equations and inequalities in algebra.
Example
Consider the equation \( 2x = 14 \).
To find the value of \( x \), utilize the Division Postulate to divide both sides of the equation by 2:
\[ x = 7 \]
Angles Context
In geometry, especially when dealing with angles, the Multiplication and Division Postulates enable us to determine the measure of unknown angles given the measures of others.
Example
If \( m\angle ABC = \frac{1}{3} \times m\angle DEF \) and \( m\angle DEF = 90^\circ \), determine the measure of \( m\angle ABC \).
Applying the Multiplication Postulate:
\[ m\angle ABC = \frac{1}{3} \times 90^\circ \]
\[ m\angle ABC = 30^\circ \]
Segments Context
In the scenario of segment measures in geometry, the Multiplication and Division Postulates are instrumental in finding unknown segment lengths given the lengths of other segments.
Example
Given that segment \( AB = \frac{1}{2} \times CD \) and \( CD = 10 \) cm, find the length of segment \( AB \).
Employing the Multiplication Postulate:
\[ AB = \frac{1}{2} \times 10 \text{ cm} \]
\[ AB = 5 \text{ cm} \]