Distributive Property in Geometry - Pre Algebra

1. Two rectangles share a height $h$, with bases 8 and 5. What single term gives their combined area?

   $13h$. $8h + 5h = h(8 + 5) = 13h$.

2. How does splitting a rectangle into two smaller rectangles model $a(b + c) = ab + ac$?

   A rectangle of width $a$ and length $b + c$ splits into rectangles of areas $ab$ and $ac$, which add to the whole area $a(b + c)$.

3. A rectangle 5 units wide is split into parts 7 and 3 units long. What equation shows the distributive property for its area?

   $5(7 + 3) = 5 \cdot 7 + 5 \cdot 3 = 35 + 15 = 50$ square units.

4. A square has side length $s + 1$. Its perimeter $4(s + 1)$ equals what expanded expression?

   $4s + 4$.

5. Expand the area $(x + 2)(x + 3)$ of a rectangle.

   $x^2 + 5x + 6$. Distribute: $x^2 + 3x + 2x + 6$, then combine like terms.

6. A rectangle has length $2x + 5$ and width 3. What is its perimeter, simplified?

   $4x + 16$. $2(2x + 5) + 2(3) = 4x + 10 + 6$.

7. A triangle has base $4m + 2$ and height 6. What is its area as an expanded expression?

   $12m + 6$. $\frac{1}{2}(6)(4m + 2) = 3(4m + 2) = 12m + 6$.

8. A rectangle has area $6x + 18$ and width 6. What expression gives its length?

   $x + 3$. Factoring gives $6x + 18 = 6(x + 3)$.

9. Using the distributive property, what factored form is equivalent to the rectangle perimeter $2l + 2w$?

   $2(l + w)$. Factoring out the common 2 gives twice the sum of length and width.

10. A rectangle has length $x + 4$ and width $x$. What is its area, expanded?

    $x^2 + 4x$. $x(x + 4) = x^2 + 4x$.

11. Use the distributive property to rewrite the trapezoid area $A = \frac{1}{2}h(b_1 + b_2)$ without parentheses.

    $A = \frac{1}{2}hb_1 + \frac{1}{2}hb_2$, the sum of two triangle areas with height $h$.

12. A rectangle has width 4 and length $x + 3$. What expanded expression gives its area?

    $4x + 12$. By the distributive property, $4(x + 3) = 4x + 12$.

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