Find the area of the trapezium ABCD in which AB//DC, AB = 18 cm, `angleB = angleC = 90^(@)`, CD = 12 cm and AD = 10 cm.

Answer

To find the area of trapezium ABCD, we will follow these steps:

### Step 1: Understand the given data
We have the following information:
- AB is parallel to CD (AB // CD)
- Length of AB = 18 cm
- Length of CD = 12 cm
- Angle B = Angle C = 90° (which means both sides BC and AD are perpendicular to AB and CD)
- Length of AD = 10 cm

### Step 2: Draw the trapezium
Draw trapezium ABCD where:
- AB is the top base (18 cm)
- CD is the bottom base (12 cm)
- Points B and C are at right angles to AB and CD.

### Step 3: Identify the height of the trapezium
Since angles B and C are both 90°, the height (h) of the trapezium is the vertical distance between the two parallel sides, which can be found by calculating the length of segment AM (the horizontal distance from point A to point M, where M is the vertical projection of point D onto line AB).

### Step 4: Calculate the horizontal distance AM
Since AB = 18 cm and CD = 12 cm, the horizontal distance AM can be calculated as:
\[ AM = AB - CD = 18 \, \text{cm} - 12 \, \text{cm} = 6 \, \text{cm} \]

### Step 5: Use the Pythagorean theorem to find the height
In triangle AMD, we can apply the Pythagorean theorem:
- AD is the hypotenuse = 10 cm
- AM is one leg = 6 cm
- Let the height (h) be the other leg.

Using the Pythagorean theorem:
\[ AD^2 = AM^2 + h^2 \]
\[ 10^2 = 6^2 + h^2 \]
\[ 100 = 36 + h^2 \]
\[ h^2 = 100 - 36 \]
\[ h^2 = 64 \]
\[ h = \sqrt{64} = 8 \, \text{cm} \]

### Step 6: Calculate the area of the trapezium
The area \( A \) of a trapezium is given by the formula:
\[ A = \frac{1}{2} \times (AB + CD) \times h \]
Substituting the values:
\[ A = \frac{1}{2} \times (18 \, \text{cm} + 12 \, \text{cm}) \times 8 \, \text{cm} \]
\[ A = \frac{1}{2} \times 30 \, \text{cm} \times 8 \, \text{cm} \]
\[ A = 15 \, \text{cm} \times 8 \, \text{cm} \]
\[ A = 120 \, \text{cm}^2 \]

### Final Answer
The area of trapezium ABCD is \( 120 \, \text{cm}^2 \).

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