ABCD is a trapezium such that AB, DC is parallel and BC is perpendicular to them. If angle (DAB) = `45^(@)` , BC = 2 cm and CD = 3 cm, then find the length of AB?

Options

A. 6 cm
B. 4 cm
C. 3 cm
D. 5 cm

Answer

To find the length of AB in trapezium ABCD, where AB is parallel to CD, BC is perpendicular to both AB and CD, and angle DAB is 45 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Trapezium Configuration**:
- Given trapezium ABCD, with AB || CD and BC ⊥ AB, CD.
- BC = 2 cm, CD = 3 cm, and angle DAB = 45°.

2. **Draw the Trapezium**:
- Draw a trapezium ABCD such that AB is at the top, CD is at the bottom, and BC is vertical.
- Label the points accordingly.

3. **Identify the Right Triangle**:
- In triangle DAB, we have:
- Angle DAB = 45°
- BC (height) = 2 cm
- CD (base) = 3 cm

4. **Use Trigonometric Ratios**:
- Since angle DAB is 45°, we can use the properties of a 45-45-90 triangle.
- In a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse is \( \sqrt{2} \) times the length of each leg.

5. **Calculate the Length of AB**:
- Let AB = x.
- In triangle DAB:
- Using the sine function: \( \sin(45°) = \frac{BC}{AD} \)
- Here, \( AD = AB + CD = x + 3 \)
- Therefore, \( \sin(45°) = \frac{2}{x + 3} \)
- Since \( \sin(45°) = \frac{1}{\sqrt{2}} \), we can set up the equation:
\[ \frac{1}{\sqrt{2}} = \frac{2}{x + 3} \]

6. **Cross-Multiply and Solve for x**:
- Cross-multiplying gives:
\[ x + 3 = 2\sqrt{2} \]
- Rearranging gives:
\[ x = 2\sqrt{2} - 3 \]

7. **Calculate the Numerical Value**:
- Using \( \sqrt{2} \approx 1.414 \):
\[ x \approx 2 \times 1.414 - 3 \approx 2.828 - 3 \approx -0.172 \]
- Since length cannot be negative, we need to reevaluate our assumptions or calculations.

8. **Reassess the Geometry**:
- Since we have a trapezium, we should consider the trapezium's properties and the relationship between the sides.
- The correct approach is to use the Pythagorean theorem in triangle DAB:
\[ AB = CD + BC \cdot \tan(45°) \]
- Since \( \tan(45°) = 1 \):
\[ AB = 3 + 2 = 5 \text{ cm} \]

### Final Answer:
The length of AB is **5 cm**.