Let's break down the problem from the image you've uploaded. The question asks to calculate the following:

Given Information:

• A trapezoid $ABCD$ with:
  • $AD = 9 \, \text{cm}$
  • $AB = 14 \, \text{cm}$
  • $DE = 24 \, \text{cm}$
  • $BD = 12 \, \text{cm}$

To Calculate:

a. The area of triangle $ABD$.
b. The area of triangle $BCD$.
c. The area of trapezoid $ABCD$.

I will start by calculating the areas step by step.

a. Area of Triangle ABD:

To find the area of triangle $ABD$, we can use the formula for the area of a triangle: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base is $AB = 14 \, \text{cm}$ and the height is $AD = 9 \, \text{cm}$. $\text{Area of } ABD = \frac{1}{2} \times 14 \times 9 = 63 \, \text{cm}^2$

b. Area of Triangle BCD:

For triangle $BCD$, we use the same formula. The base is $BD = 12 \, \text{cm}$ and the height is the perpendicular distance between lines $DE$ and $BD$, which is given as $AD = 9 \, \text{cm}$ because it's a trapezoid and the heights are the same. $\text{Area of } BCD = \frac{1}{2} \times 12 \times 9 = 54 \, \text{cm}^2$

c. Area of Trapezoid ABCD:

The area of the trapezoid can be found using the formula for the area of a trapezoid: $\text{Area of trapezoid} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{height}$ Here, the two bases are $AB = 14 \, \text{cm}$ and $DE = 24 \, \text{cm}$, and the height is $AD = 9 \, \text{cm}$. $\text{Area of trapezoid ABCD} = \frac{1}{2} \times (14 + 24) \times 9 = \frac{1}{2} \times 38 \times 9 = 171 \, \text{cm}^2$

Summary:

• a. Area of triangle ABD = $63 \, \text{cm}^2$
• b. Area of triangle BCD = $54 \, \text{cm}^2$
• c. Area of trapezoid ABCD = $171 \, \text{cm}^2$

Would you like further clarifications or details?

Related Questions:

1. How do we calculate the area of a triangle using Heron's formula?
2. What is the significance of height in calculating the area of trapezoids?
3. How can you calculate the area of an irregular shape using decomposition?
4. Can we calculate the perimeter of the trapezoid ABCD?
5. How do you apply Pythagoras' theorem to find the height of a triangle in certain cases?

Tip:

Always identify and label known dimensions clearly when solving for areas, especially in complex shapes like trapezoids and triangles.