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A diagram shows a large rectangle divided into four smaller shapes: VCAZ, ZADY, CWBA, and ABXD. Determine the area of each smaller shape.

Mathematics

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3.1.1 Area of VCAZ

  1. Identify the shape and its dimensions.

    VCAZ is a square with side length VC=xVC = x units.

  2. Apply the formula for the area of a square.

    Area of VCAZ =side×side=x×x=x2= \text{side} \times \text{side} = x \times x = x^2 units2^2.
    The area of VCAZ is x2 Units2\boxed{x^2 \text{ Units}^2}.

    3.1.2 Area of ZADY

  3. Identify the shape and its dimensions.

    ZADY is a rectangle. Its length is ZA=xZA = x units. Its width is ZY=2ZY = 2 units.

  4. Apply the formula for the area of a rectangle.

    Area of ZADY =length×width=ZA×ZY=x×2=2x= \text{length} \times \text{width} = ZA \times ZY = x \times 2 = 2x units2^2.
    The area of ZADY is 2x Units2\boxed{2x \text{ Units}^2}.

    3.1.3 Area of CWBA

  5. Identify the shape and its dimensions.

    CWBA is a rectangle. Its length is CW=2CW = 2 units. Its width is CBCB. From the diagram, CB=VACB = VA. Since VCAZ is a square, VA=VC=xVA = VC = x units. So, CB=xCB = x units.

  6. Apply the formula for the area of a rectangle.

    Area of CWBA =length×width=CW×CB=2×x=2x= \text{length} \times \text{width} = CW \times CB = 2 \times x = 2x units2^2.
    The area of CWBA is 2x Units2\boxed{2x \text{ Units}^2}.

    3.1.4 Area of ABXD

  7. Identify the shape and its dimensions.

    ABXD is a square with side length AB=2AB = 2 units.

  8. Apply the formula for the area of a square.

    Area of ABXD =side×side=2×2=4= \text{side} \times \text{side} = 2 \times 2 = 4 units2^2.
    The area of ABXD is 4 Units2\boxed{4 \text{ Units}^2}.

    3.1.5 (x+2)(x+2)=(x+2)(x+2)=

  9. Expand the product.

    (x+2)(x+2)=x(x+2)+2(x+2)(x+2)(x+2) = x(x+2) + 2(x+2)
  10. Distribute and simplify.

    =x2+2x+2x+4= x^2 + 2x + 2x + 4 =x2+4x+4= \boxed{x^2 + 4x + 4}
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