https://file.aiquickdraw.com/mathbot/file/1728319859016ugkk9wq2.jpg

To solve for the area of trapezium M, we need to break down the problem based on the information provided.

### Step 1: Understand the structure of the diagram
- The total height of the larger triangle (including the trapezium and the small triangle) is 45 cm.
- The height of the small triangle, which is shaded pink, is 9 cm, and its area is given as 20 cm².

### Step 2: Calculate the base of the small triangle
We know the area of a triangle is given by:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the known area and height for the small triangle:

\[
20 = \frac{1}{2} \times \text{base} \times 9
\]

Solving for the base:

\[
20 = 4.5 \times \text{base}
\]
\[
\text{base} = \frac{20}{4.5} \approx 4.44 \, \text{cm}
\]

### Step 3: Calculate the base of the large triangle
The total height of the larger triangle is 45 cm, so we now use a similar approach to find the base of the larger triangle. Using the same ratio for base and height between the two triangles:

\[
\frac{\text{base of large triangle}}{\text{base of small triangle}} = \frac{\text{height of large triangle}}{\text{height of small triangle}} = \frac{45}{9} = 5
\]

Thus, the base of the large triangle is:

\[
\text{base of large triangle} = 5 \times 4.44 \approx 22.22 \, \text{cm}
\]

### Step 4: Calculate the area of the large triangle
Now we calculate the area of the large triangle using its base and height:

\[
\text{Area of large triangle} = \frac{1}{2} \times \text{base of large triangle} \times \text{height of large triangle}
\]
\[
\text{Area of large triangle} = \frac{1}{2} \times 22.22 \times 45 \approx 500 \, \text{cm}^2
\]

### Step 5: Calculate the area of the trapezium
Finally, the area of trapezium M is the difference between the area of the large triangle and the small triangle:

\[
\text{Area of trapezium M} = \text{Area of large triangle} - \text{Area of small triangle}
\]
\[
\text{Area of trapezium M} = 500 - 20 = 480 \, \text{cm}^2
\]

Thus, the area of trapezium M is **480 cm²**.

Do you have any questions or need more details? 

Here are five related questions you might explore:
1. How can we find the perimeter of trapezium M?
2. What would be the area if the height of the large triangle were increased by 10 cm?
3. How does the ratio of areas between the small triangle and the large triangle affect the result?
4. Can you calculate the area of the trapezium using trapezium-specific formulas instead of subtraction?
5. How would the area change if the base of the small triangle was different?

**Tip:** When solving problems with composite shapes, consider calculating the areas of individual parts and then combine or subtract as needed.

Work out the area of trapezium M. If your answer is a decimal, give it to 1 d.p.

Find the area of trapezium M by calculating the areas of the small and large triangles. This step-by-step guide explains how to use proportional relationships and area formulas to solve this geometry problem. Suitable for Grades 8-10.

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