Geometry

1. The area of a triangle is $A = \frac{1}{2} \times b \times h$:

$A = \frac{1}{2} \times 14 \times 9 = \frac{126}{2} = 63 \text{ cm}^2$

2. The area of the parallelogram is $A = b \times h$:

$A = 12 \times 7 = 84 \text{ in}^2$

A diagonal divides the parallelogram into two congruent triangles, so each triangle has half the area:

$A_{\text{triangle}} = \frac{84}{2} = 42 \text{ in}^2$

3. The volume of a right rectangular prism is $V = l \times w \times h$:

$V = 8 \times 5 \times 3 = 120 \text{ cm}^3$

Since each unit cube has a volume of $1 \text{ cm}^3$, exactly $120$ unit cubes fit inside the prism.

4. Area of the rectangle:

$A_{\text{rect}} = 10 \times 6 = 60 \text{ m}^2$

Area of the right triangle:

$A_{\text{tri}} = \frac{1}{2} \times 6 \times 4 = 12 \text{ m}^2$

Total area of the polygon:

$A_{\text{total}} = 60 + 12 = 72 \text{ m}^2$

5. The net of the rectangular prism consists of six rectangles (three pairs of congruent faces):

• Two faces of $5 \times 3 = 15 \text{ cm}^2$ (top and bottom)
• Two faces of $5 \times 2 = 10 \text{ cm}^2$ (front and back)
• Two faces of $3 \times 2 = 6 \text{ cm}^2$ (left and right)

Total surface area:

$SA = 2(15) + 2(10) + 2(6) = 30 + 20 + 12 = 62 \text{ cm}^2$