Question

competency 002—the teacher understands number operations and computational algorithms. 2. rectangle i has dimensions a and b, and rectangle ii has dimensions a - 2 and b + 2, where a > 2 and b > 0. which of the following must be true? a. the area of rectangle i is less than the area of rectangle ii. b. the area of rectangle i is greater than the area of rectangle ii. c. the perimeter of rectangle i is less than the perimeter of rectangle ii. d. the perimeter of rectangle i is equal to the perimeter of rectangle ii.

Explanation:

Step1: Calculate Area of Rectangle I

The area of a rectangle is given by the formula \( A = \text{length} \times \text{width} \). For Rectangle I with dimensions \( a \) and \( b \), the area \( A_1 = a \times b = ab \).

Step2: Calculate Area of Rectangle II

For Rectangle II with dimensions \( a - 2 \) and \( b + 2 \), the area \( A_2=(a - 2)(b + 2) \). We expand this using the distributive property (FOIL method):
\[

$$\begin{align*} A_2&=(a - 2)(b + 2)\\ &=ab+2a-2b - 4 \end{align*}$$

\]

Step3: Compare \( A_1 \) and \( A_2 \)

Now, we find the difference between \( A_2 \) and \( A_1 \):
\[

$$\begin{align*} A_2 - A_1&=(ab + 2a-2b - 4)-ab\\ &=2a-2b - 4\\ &=2(a - b - 2) \end{align*}$$

\]
We don't know the relationship between \( a \) and \( b \) to determine if \( A_2>A_1 \) or \( A_2 < A_1 \), so we can't conclude about the area.

Step4: Calculate Perimeter of Rectangle I

The perimeter of a rectangle is given by \( P = 2(\text{length}+\text{width}) \). For Rectangle I, \( P_1 = 2(a + b)=2a + 2b \).

Step5: Calculate Perimeter of Rectangle II

For Rectangle II, \( P_2 = 2((a - 2)+(b + 2)) \). Simplify the expression inside the parentheses:
\[

$$\begin{align*} (a - 2)+(b + 2)&=a - 2 + b + 2\\ &=a + b \end{align*}$$

\]
So, \( P_2 = 2(a + b)=2a + 2b \).

Step6: Compare Perimeters

Since \( P_1 = 2a + 2b \) and \( P_2 = 2a + 2b \), we have \( P_1 = P_2 \).

Answer:

D. The perimeter of rectangle I is equal to the perimeter of rectangle II.