Question

shape 1 area 24 cm_____
shape 2 area 80 cm_____
shape 3 area 16 cm_____
= 120 cm²
(image of a composite figure with a triangle on top, a larger rectangle, and a smaller rectangle; dimensions: triangle height 6 cm, larger rectangle height 10 cm, total width 12 cm, smaller rectangle width 4 cm, height 6 cm; two ? for unknown lengths)

Explanation:

To solve for the unknowns (the base of the triangle, the height of the small rectangle), we analyze each shape:

Step 1: Find the base of the triangle (Shape 1)

Shape 1 is a triangle with area \( 24 \, \text{cm}^2 \) and height \( 6 \, \text{cm} \). The formula for the area of a triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Rearranging to solve for the base:

$$ \text{base} = \frac{2A}{\text{height}} $$

Substitute \( A = 24 \, \text{cm}^2 \) and \( \text{height} = 6 \, \text{cm} \):

$$ \text{base} = \frac{2 \times 24}{6} = \frac{48}{6} = 8 \, \text{cm} $$

Step 2: Find the height of the small rectangle (Shape 3)

Shape 3 is a rectangle with area \( 16 \, \text{cm}^2 \) and width \( 4 \, \text{cm} \). The formula for the area of a rectangle is \( A = \text{width} \times \text{height} \). Rearranging to solve for the height:

$$ \text{height} = \frac{A}{\text{width}} $$

Substitute \( A = 16 \, \text{cm}^2 \) and \( \text{width} = 4 \, \text{cm} \):

$$ \text{height} = \frac{16}{4} = 4 \, \text{cm} $$

Final Answers:

• The base of the triangle (the "?" above the large rectangle) is \( \boldsymbol{8 \, \text{cm}} \).
• The height of the small rectangle (the "?" on the left) is \( \boldsymbol{4 \, \text{cm}} \).

Answer:

To solve for the unknowns (the base of the triangle, the height of the small rectangle), we analyze each shape:

Step 1: Find the base of the triangle (Shape 1)

Shape 1 is a triangle with area \( 24 \, \text{cm}^2 \) and height \( 6 \, \text{cm} \). The formula for the area of a triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Rearranging to solve for the base:

$$ \text{base} = \frac{2A}{\text{height}} $$

Substitute \( A = 24 \, \text{cm}^2 \) and \( \text{height} = 6 \, \text{cm} \):

$$ \text{base} = \frac{2 \times 24}{6} = \frac{48}{6} = 8 \, \text{cm} $$

Step 2: Find the height of the small rectangle (Shape 3)

Shape 3 is a rectangle with area \( 16 \, \text{cm}^2 \) and width \( 4 \, \text{cm} \). The formula for the area of a rectangle is \( A = \text{width} \times \text{height} \). Rearranging to solve for the height:

$$ \text{height} = \frac{A}{\text{width}} $$

Substitute \( A = 16 \, \text{cm}^2 \) and \( \text{width} = 4 \, \text{cm} \):

$$ \text{height} = \frac{16}{4} = 4 \, \text{cm} $$

Final Answers:

• The base of the triangle (the "?" above the large rectangle) is \( \boldsymbol{8 \, \text{cm}} \).
• The height of the small rectangle (the "?" on the left) is \( \boldsymbol{4 \, \text{cm}} \).