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To solve the given problem about finding the area of a trapezoid using two different methods, let’s analyze the shape and approach it step by step.

### Part (a): **Two methods to find the area of the trapezoid**
We are given the trapezoid's dimensions:
- The two parallel sides (bases) are 7 cm and 10 cm.
- The height (perpendicular distance between the two bases) is 8 cm.
- The length between the two bases is 6 cm, forming the non-parallel sides of the trapezoid.

#### **Method 1: Using the trapezoid area formula**
The area \(A\) of a trapezoid is given by the formula:
\[
A = \frac{1}{2} \times ( \text{Base}_1 + \text{Base}_2 ) \times \text{Height}
\]
- Base 1 = 7 cm
- Base 2 = 10 cm
- Height = 8 cm

Substituting the values:
\[
A = \frac{1}{2} \times (7 + 10) \times 8 = \frac{1}{2} \times 17 \times 8 = 68 \, \text{cm}^2
\]

#### **Method 2: Dividing the trapezoid into simpler shapes**
We can divide the trapezoid into a rectangle and two right triangles.
- The middle portion is a rectangle with width 6 cm and height 8 cm.
- Two right triangles on the sides, where the base lengths are \((10 - 6) / 2 = 2 \, \text{cm}\) and \((7 - 6) / 2 = 0.5 \, \text{cm}\), both having a height of 8 cm.

**Area of the rectangle**:
\[
A_{\text{rectangle}} = \text{width} \times \text{height} = 6 \times 8 = 48 \, \text{cm}^2
\]

**Area of the left triangle**:
\[
A_{\text{left triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 0.5 \times 8 = 2 \, \text{cm}^2
\]

**Area of the right triangle**:
\[
A_{\text{right triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 8 = 8 \, \text{cm}^2
\]

The total area is:
\[
A_{\text{total}} = A_{\text{rectangle}} + A_{\text{left triangle}} + A_{\text{right triangle}} = 48 + 2 + 8 = 68 \, \text{cm}^2
\]

### Part (b): **Show that both methods give the same area**
Both methods give the same area, \( 68 \, \text{cm}^2 \), which confirms that the calculations are correct and consistent.

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Would you like a further breakdown of any step? Feel free to ask.

Here are some additional related questions:

1. How would you find the perimeter of this trapezoid?
2. How can you use the Pythagorean theorem to verify the sides of the triangles?
3. What is the formula for the area of a triangle and why does it work here?
4. How would the method change if the trapezoid were isosceles?
5. How does changing the height affect the area of the trapezoid?

**Tip:** Always check if the shape can be divided into simpler geometric figures to simplify area calculations!

Describe two different methods you could use to find the area of the trapezoid, and show that both methods give the same area. The trapezoid has base lengths of 7 cm and 10 cm, with a height of 8 cm and a middle section of 6 cm.

Learn how to calculate the area of a trapezoid with bases 7 cm and 10 cm, and a height of 8 cm. Two methods are used: the trapezoid area formula and breaking the shape into simpler figures (a rectangle and triangles). This step-by-step guide is perfect for students in Grades 6-8.

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