The Distributive Property And Rectangle Area

🔑 Key Concepts

  • The total width of the combined rectangle is \(3x+2\).
  • The area of a rectangle is found using \(A=lw\).
  • The expression \(9x+6\) can be factored as \(3(3x+2)\).
  • Equivalent expressions can reveal an unknown geometric dimension.
  • The common factor of the area terms represents the shared height.
  • A solution should be checked by distributing to recover the original area expression.

✏️ Worked Example — Find the Missing Rectangle Dimension

🧠 Math Vocabulary

Area

The amount of space inside a two-dimensional figure, measured in square units.

Dimension

A measurable length, width, or height of a geometric figure.

Equivalent Expressions

Expressions that have the same value for every permitted value of their variables.

Factor

A number or expression multiplied by another number or expression.

Greatest Common Factor

The greatest factor shared by every term in an expression.

Distributive Property

A property connecting multiplication and addition, such as \(a(b+c)=ab+ac\).

💡 Main Idea

Algebraic area expressions can reveal missing dimensions of geometric figures. The combined rectangle has a total width of \(3x+2\) and an area of \(9x+6\). Factoring the area gives \(3(3x+2)\). Because the factor \(3x+2\) matches the rectangle’s width, the remaining factor, \(3\), represents its height. Therefore, the length of \(\overline{AB}\) is \(3\) units.

📚 What You Should Already Know

Students should know how to find the area of a rectangle, add algebraic side lengths, identify common factors, apply the distributive property, and recognize equivalent expressions. Students should also understand that adjacent rectangles sharing the same height can be treated as one larger rectangle.

🚀 What Comes Next

Students can extend this reasoning to composite figures, missing dimensions, perimeter expressions, and equations involving geometric measurements. Factoring and the distributive property will also support later work with linear equations, polynomial expressions, and algebraic formulas.

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