Calculate Rectangle Dimensions: From 8cm × 12cm to 200cm Perimeter

To solve this problem, let's first verify the understanding of similar rectangles:

• Step 1: Compute the perimeter of the first rectangle.

  The formula for the perimeter is $P = 2 \times (\text{length} + \text{width})$. Thus, for the first rectangle:

  $P = 2 \times (8\, \text{cm} + 12\, \text{cm}) = 2 \times 20\, \text{cm} = 40\, \text{cm}$.

• Step 2: Establish the relationship of sides between similar rectangles.

  Since the perimeters differ, the corresponding side lengths differ while maintaining proportionality across the rectangles.

  If the known sides are $8 \, \text{cm}$ and $12 \, \text{cm}$, and the second rectangle has a perimeter of $200 \, \text{cm}$:

• Step 3: Calculate dimensions of the second rectangle.

  Since both rectangles are similar, their corresponding side lengths are scaled versions of each other. Let's denote the side lengths of the second rectangle as $s_1$ and $s_2$.

  We have the equation for the perimeter:

  $2 \times (s_1 + s_2) = 200$.

  This simplifies to:

  $s_1 + s_2 = 100$. (1)

• Step 4: Find the scaling factor based on similarity of rectangles.

  The sides are proportional: $\frac{8}{12} = \frac{s_1}{s_2}$.

  Solving this proportion, we get:

  $12s_1 = 8s_2$ which simplifies to $\frac{s_1}{s_2} = \frac{2}{3}$.

  Now substitute $s_1 = \frac{2}{3}s_2$ into equation (1):

  $\frac{2}{3}s_2 + s_2 = 100$

  $\frac{5}{3}s_2 = 100$

  Solving for $s_2$, we multiply both sides by $\frac{3}{5}$:

  $s_2 = 60 \, \text{cm}$.

  Substituting $s_2$ back into the equation for $s_1$:

  $s_1 = 100 - 60 = 40 \, \text{cm}$.

Therefore, the lengths of the second rectangle's sides are 60 cm and 40 cm.