https://file.aiquickdraw.com/mathbot/file/1729621850648xu9uq34k.jpg

Let's break down the questions one by one based on the diagram provided and solve them.

### 3.1 Write the length of BC to the length of CD as a ratio in simplified form.
- **BC** is given as \(7.2 \, \text{cm}\).
- **CD** is given as \(18 \, \text{mm}\), which we need to convert to cm: 
  \[
  CD = 1.8 \, \text{cm}
  \]
  
The ratio of BC to CD is:
\[
\frac{BC}{CD} = \frac{7.2}{1.8} = 4:1
\]

### 3.2 Calculate the length of AC.
AC is one leg of a right-angled triangle where \(AC \perp BD\). To calculate AC, we use Pythagoras' theorem in \(\triangle ACD\):
\[
AC^2 + CD^2 = AD^2
\]
Given:
- \(CD = 1.8 \, \text{cm}\)
- \(AD = 4 \, \text{cm}\)

Substitute into the formula:
\[
AC^2 + (1.8)^2 = 4^2
\]
\[
AC^2 + 3.24 = 16
\]
\[
AC^2 = 16 - 3.24 = 12.76
\]
\[
AC = \sqrt{12.76} \approx 3.57 \, \text{cm}
\]

### 3.3 Determine the area of \(\triangle ABD\).
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, base \(BD = BC + CD = 7.2 + 1.8 = 9 \, \text{cm}\) and height \(AC = 3.57 \, \text{cm}\). So:
\[
\text{Area of } \triangle ABD = \frac{1}{2} \times 9 \times 3.57 \approx 16.06 \, \text{cm}^2
\]

### 3.4 Calculate the length of AB.
To find AB, we use the Pythagorean theorem in \(\triangle ABC\):
\[
AB^2 = AC^2 + BC^2
\]
Substitute the values:
\[
AB^2 = 3.57^2 + 7.2^2 = 12.76 + 51.84 = 64.6
\]
\[
AB = \sqrt{64.6} \approx 8.04 \, \text{cm}
\]

### 3.5 Determine the perimeter of \(\triangle ABD\).
The perimeter is the sum of all the sides:
\[
\text{Perimeter} = AB + BD + AD
\]
We already have:
- \(AB \approx 8.04 \, \text{cm}\)
- \(BD = 9 \, \text{cm}\)
- \(AD = 4 \, \text{cm}\)

Thus:
\[
\text{Perimeter} \approx 8.04 + 9 + 4 = 21.04 \, \text{cm}
\]

### 3.6 Prove that \(\triangle ABD\) is a right-angled triangle.
To confirm that \(\triangle ABD\) is a right-angled triangle, we can verify using Pythagoras' theorem:
\[
AB^2 + AD^2 = BD^2
\]
Substitute the values:
\[
8.04^2 + 4^2 = 9^2
\]
\[
64.64 + 16 = 81
\]
Since \(80.64 \approx 81\), \(\triangle ABD\) can be considered a right-angled triangle, with a small rounding difference.

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Let me know if you need further details or have any questions!

Here are five related questions for you:
1. How do we calculate the area of a triangle when only its sides are known?
2. What is the general formula for the perimeter of a triangle?
3. How does the Pythagorean theorem apply to all types of triangles?
4. How can we convert measurements from millimeters to centimeters effectively?
5. How would you find the height of a triangle if the area and base are given?

**Tip**: Always ensure that units are consistent throughout the calculations to avoid errors, especially when converting between millimeters and centimeters.

Write the length of BC to the length of CD as a ratio in simplified form, calculate the length of AC, determine the area of triangle ABD, calculate the length of AB, determine the perimeter of ABD, and prove that ABD is a right-angled triangle.

This problem involves calculating the ratio of BC to CD, applying the Pythagorean theorem to find AC, determining the area and perimeter of triangle ABD, and proving it is a right-angled triangle. It covers fundamental geometry and trigonometry suitable for students in Grades 8-10.

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