To solve the problem of finding the area of the shaded triangle region $\triangle AEF$, let us break it into steps:

Step 1: Understand the given dimensions

• Rectangle $ABCD$ has:

  • Height = $15 \, \text{cm}$
  • Width = $24 \, \text{cm}$

• Segment $DE = 8 \, \text{cm}$, so $EC = 15 - 8 = 7 \, \text{cm}$.

• Segment $DF = 12 \, \text{cm}$, so $FC = 24 - 12 = 12 \, \text{cm}$.

• $\triangle AEF$ is the triangle we need to find the area of, and it is located within the rectangle.

Step 2: Formula for the area of a triangle

The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

For $\triangle AEF$:

• Base = $DF = 12 \, \text{cm}$
• Height = $DE = 8 \, \text{cm}$

Step 3: Compute the area of $\triangle AEF$

Substitute the values into the formula: $\text{Area of } \triangle AEF = \frac{1}{2} \times 12 \, \text{cm} \times 8 \, \text{cm}$ $\text{Area of } \triangle AEF = \frac{1}{2} \times 96 \, \text{cm}^2$ $\text{Area of } \triangle AEF = 48 \, \text{cm}^2$

Step 4: Subtract the triangle's area from the rectangle's area

The area of the rectangle $ABCD$ is: $\text{Area of rectangle } ABCD = \text{Length} \times \text{Width} = 24 \, \text{cm} \times 15 \, \text{cm} = 360 \, \text{cm}^2$

The area of the shaded region is the difference: $\text{Shaded area} = \text{Area of rectangle } ABCD - \text{Area of } \triangle AEF$ $\text{Shaded area} = 360 \, \text{cm}^2 - 48 \, \text{cm}^2$ $\text{Shaded area} = 312 \, \text{cm}^2$

Final Answer:

$\boxed{B) 312 \, \text{cm}^2}$Apologies for the confusion earlier. Upon reviewing the problem carefully, the shaded region is exactly the triangle $\triangle AEF$. Let me correct the solution, focusing only on finding the area of $\triangle AEF$. Here's the correct step-by-step solution:

Step 1: Formula for the area of a triangle

The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

For $\triangle AEF$:

• Base = $DF = 12 \, \text{cm}$
• Height = $DE = 8 \, \text{cm}$ (perpendicular from point $E$ to base $DF$).

Step 2: Compute the area of $\triangle AEF$

Substitute the values into the formula: $\text{Area of } \triangle AEF = \frac{1}{2} \times 12 \, \text{cm} \times 8 \, \text{cm}$ $\text{Area of } \triangle AEF = \frac{1}{2} \times 96 \, \text{cm}^2$ $\text{Area of } \triangle AEF = 48 \, \text{cm}^2$

Final Answer:

$\boxed{D) 48 \, \text{cm}^2}$

Let me know if you'd like additional details or clarifications!
Here are 5 related questions to deepen your understanding:

1. How do you find the height of a triangle if it's not explicitly given?
2. Can the area of a rectangle be used to approximate areas of other shapes within it?
3. How do the dimensions of a rectangle affect the areas of its internal triangles?
4. What is the significance of perpendicular heights in calculating triangle areas?
5. Can you calculate the area of irregular polygons by dividing them into triangles?

Tip: Always confirm the perpendicular height of a triangle when calculating its area to avoid errors.