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To solve for the perimeter of the trapezium \( ABCD \), we need to calculate the length of the side \( AB \) because we already know the lengths of \( AD = 15 \, \text{cm} \), \( DC = 14 \, \text{cm} \), and we’ll use \( BC \) once we find \( AB \).

Given information:
- \( AD = 15 \, \text{cm} \)
- \( DC = 14 \, \text{cm} \)
- Area of trapezium \( ABCD = 360 \, \text{cm}^2 \)
- \( DAB \) and \( ADC \) are right angles, so \( AD \) is perpendicular to \( AB \) and \( DC \).

### Step 1: Set up the area formula for the trapezium
The area \( A \) of a trapezium with parallel sides \( a \) and \( b \), and height \( h \), is given by:
\[
A = \frac{1}{2} \times (a + b) \times h
\]
In this problem:
- \( a = DC = 14 \, \text{cm} \)
- \( b = AB \)
- \( h = AD = 15 \, \text{cm} \)
- \( A = 360 \, \text{cm}^2 \)

Substituting these values into the area formula, we get:
\[
360 = \frac{1}{2} \times (14 + AB) \times 15
\]

### Step 2: Solve for \( AB \)
Rearrange the equation:
\[
360 = \frac{1}{2} \times (14 + AB) \times 15
\]
Multiply both sides by 2:
\[
720 = (14 + AB) \times 15
\]
Now, divide by 15:
\[
48 = 14 + AB
\]
Subtract 14 from both sides:
\[
AB = 34 \, \text{cm}
\]

### Step 3: Calculate the length of \( BC \)
Now, since \( ABCD \) is a trapezium with right angles at \( DAB \) and \( ADC \), we can calculate \( BC \) by recognizing that triangle \( BCD \) is a right triangle with:
- \( BD = AB - DC = 34 - 14 = 20 \, \text{cm} \)
- \( CD = 14 \, \text{cm} \)

Using the Pythagorean theorem for triangle \( BCD \):
\[
BC = \sqrt{BD^2 + CD^2}
\]
\[
BC = \sqrt{20^2 + 14^2}
\]
\[
BC = \sqrt{400 + 196}
\]
\[
BC = \sqrt{596}
\]
\[
BC \approx 24.4 \, \text{cm}
\]

### Step 4: Calculate the perimeter of the trapezium
Now we can find the perimeter by adding the lengths of all sides:
\[
\text{Perimeter} = AD + AB + BC + DC
\]
\[
\text{Perimeter} = 15 + 34 + 24.4 + 14
\]
\[
\text{Perimeter} \approx 87.4 \, \text{cm}
\]

### Final Answer
The perimeter of the trapezium \( ABCD \) is approximately \( 87.4 \, \text{cm} \).

Would you like more details on any step?

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Here are some related questions you might find helpful:
1. How do we derive the area formula for a trapezium?
2. Can we use another method to verify the length of \( BC \)?
3. What if the angles weren’t right angles? How would that affect the solution?
4. How would you find the area if only two sides and the height were given?
5. How does the Pythagorean theorem help in this trapezium problem?

**Tip:** Always double-check each measurement's placement in a shape, especially with right angles, to avoid calculation errors.

The diagram shows a trapezium, ABCD, with right angles at DAB and ADC. Given that AD = 15 cm, DC = 14 cm, and the area of the trapezium is 360 cm², work out the perimeter of the trapezium.

Calculate the perimeter of trapezium ABCD, with sides AD = 15 cm, DC = 14 cm, and area of 360 cm². This solution involves using trapezium area formula and the Pythagorean theorem to determine missing side lengths. Suitable for students in Grades 8-10.

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