A trapezium ABCD has side AD parallel to BC, ∠BAD = 90°, BC = 3 cm, and AD = 8 cm. If the perimeter of this trapezium is 36 cm, then its area, in square centimeters, is:

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Accepted Answer

![Screenshot 2025-04-04 131630.png](https://res.cloudinary.com/aptidude/image/upload/v1743752901/aptidude_questions/ktisuraethcnwc2ffdpq.png)To solve this problem:

1. **Perimeter Equation**:  
   The perimeter of the trapezium is given as:  
   $$
   AD + AB + BC + CD = 36
   $$  
   Substituting the known values:  
   $$
   8 + h + 3 + DC = 36
   $$  
   Simplifying:  
   $$
   h + DC = 25
   $$

2. **Pythagorean Triplet**:  
   The height $$ h $$, diagonal $$ DC $$, and the fixed segment $$ 5 $$ form a right triangle. Using the Pythagorean triplet relationship:  
   $$
   h^2 + 5^2 = DC^2
   $$

3. **Finding Dimensions**:  
   From the triplet (12, 5, 13), we find that:  
   - $$ h = 12 $$  
   - $$ DC = 13 $$

4. **Area of Trapezium**:  
   The area of a trapezium is given by:  
   $$
   	ext{Area} = \frac{1}{2} 	imes (	ext{Sum of parallel sides}) 	imes 	ext{Height}
   $$  
   Substituting the values:  
   $$
   	ext{Area} = \frac{1}{2} 	imes (8 + 3) 	imes 12 = \frac{1}{2} 	imes 11 	imes 12 = 66 \, 	ext{sq. units}
   $$

Thus, the area of the trapezium is **66 square centimeters**.