Let's solve the questions one by one based on the diagram provided:

Part (a): Find the area of the rectangle in terms of $x$ and $y$.

• The dimensions of the rectangle are given as:

  • Length = $4x$ cm
  • Width = $(3y + 2)$ cm

• The area $A_{\text{rect}}$ of a rectangle is given by: $A_{\text{rect}} = \text{Length} \times \text{Width}$ Substituting the given dimensions: $A_{\text{rect}} = 4x \times (3y + 2)$ Expanding this expression: $A_{\text{rect}} = 12xy + 8x \, \text{cm}^2$ Answer for (a): $12xy + 8x \, \text{cm}^2$

Part (b): Find the area of the triangle in terms of $x$ and $y$.

• The dimensions of the triangle are given as:

  • Base = $2x$ cm
  • Height = $(y - 1)$ cm

• The area $A_{\text{tri}}$ of a triangle is given by: $A_{\text{tri}} = \frac{1}{2} \times \text{Base} \times \text{Height}$ Substituting the given dimensions: $A_{\text{tri}} = \frac{1}{2} \times 2x \times (y - 1)$ Simplifying: $A_{\text{tri}} = x(y - 1) = xy - x \, \text{cm}^2$ Answer for (b): $xy - x \, \text{cm}^2$

Part (c): Subtract the rectangle’s area from the triangle’s area.

To subtract the area of the rectangle from the area of the triangle, we have: $\text{Difference} = A_{\text{tri}} - A_{\text{rect}}$ Substituting the expressions from parts (a) and (b): $\text{Difference} = (xy - x) - (12xy + 8x)$ Expanding and simplifying: $\text{Difference} = xy - x - 12xy - 8x$ $\text{Difference} = -11xy - 9x \, \text{cm}^2$ Answer for (c): $-11xy - 9x \, \text{cm}^2$

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Here are 5 related questions to explore further:

1. What is the perimeter of the rectangle in terms of $x$ and $y$?
2. What is the perimeter of the triangle in terms of $x$ and $y$?
3. If $x = 2$ and $y = 3$, what is the numerical value of the area of the rectangle?
4. What is the total area of the rectangle and the triangle combined in terms of $x$ and $y$?
5. How would the expressions change if the base of the triangle was $3x$ instead of $2x$?

Tip: When dealing with geometry problems, always make sure to understand the relationship between dimensions (length, width, base, height) and formulas for area and perimeter!