Question

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 sq. cm, is

Answer 1

Correct Answer: 66

Given:
Side \( BC = 3 \text{ cm} \)

• Side \( AD = 8 \text{ cm} \)
• Perimeter of the trapezium = \( 36 \text{ cm} \)
• Right angle at \( \angle BAD = 90^\circ \)

Let the other two sides be \( AB \) and \( CD \). Since \( AD \parallel BC \), the non-parallel sides are equal: \[ AB = CD \]

Step 1: Use the Perimeter Formula

Perimeter of a trapezium: \[ \text{Perimeter} = AB + BC + CD + AD \] Substituting known values: \[ 36 = AB + 3 + CD + 8 \] \[ AB + CD = 36 - (3 + 8) = 25 \] Since \( AB = CD \), we write: \[ 2 \cdot AB = 25 \Rightarrow AB = CD = \frac{25}{2} = 12.5 \text{ cm} \]

Step 2: Find Height Using Pythagoras

Since \( \angle BAD = 90^\circ \), triangle \( ABD \) is a right triangle. Use the Pythagorean Theorem: \[ BD^2 = AB^2 - AD^2 = (12.5)^2 - 8^2 = 156.25 - 64 = 92.25 \] \[ BD = \sqrt{92.25} \approx 9.61 \text{ cm} \]

Step 3: Use Area Formula for Trapezium

Formula: \[ \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \] Substituting values: \[ \text{Area} = \frac{1}{2} \times (12.5 + 12.5) \times 9.61 = \frac{1}{2} \times 25 \times 9.61 \approx 66 \text{ cm}^2 \]

Final Answer:

\[ \boxed{\text{Area} \approx 66 \text{ cm}^2} \]

Answer 2

Step 1: Apply Pythagoras Theorem

According to the right triangle in the trapezium, \[ CD^2 = y^2 + 25 \] Therefore, \[ CD = \sqrt{y^2 + 25} \]

Step 2: Use Perimeter Condition

Given the perimeter of the trapezium is 36: \[ 11 + y + \sqrt{y^2 + 25} = 36 \] Isolating the square root: \[ \sqrt{y^2 + 25} = 25 - y \]

Step 3: Square Both Sides

Square both sides to eliminate the square root: \[ y^2 + 25 = (25 - y)^2 = 625 - 50y + y^2 \] Subtract \( y^2 \) from both sides: \[ 25 = 625 - 50y \Rightarrow 50y = 600 \Rightarrow y = \frac{600}{50} = 12 \]

Step 4: Calculate Area

Area of trapezium is: \[ \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \] Here, base1 = 3, base2 = 8, height = \( y = 12 \) \[ \text{Area} = \frac{1}{2} \times (3 + 8) \times 12 = \frac{11 \cdot 12}{2} = 66 \]

Final Answer:

\[ \boxed{66 \ \text{cm}^2} \]