Question

ABCD is an isosceles trapezium with angle \( A = 45^\circ \) and the length of one of the non-parallel sides is \( 10\sqrt{2} \). The area of triangle \( ABD \) is 200 sq. units. What is the sum of the lengths of the parallel sides?

• A. 60
• B. 50
• C. 40
• D. 30

Hint:

In geometry problems involving trapezium and triangles, use trigonometry to find the height and the base, then apply the area formula to solve for unknown values.

Answer 1

The Correct Option is A

Step 1: Geometry of the Trapezium and Triangle.
We are given that \( ABCD \) is an isosceles trapezium with angles \( A = 45^\circ \) and \( D = 45^\circ \). The non-parallel sides \( AB \) and \( CD \) are equal, and each has a length of \( 10\sqrt{2} \) units.

Step 2: Use Area Formula for Triangle.
The area of triangle \( ABD \) is given as 200 square units. The area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \] For triangle \( ABD \), the base is \( BD \), and the height can be found using trigonometry.

Step 3: Finding Height Using Trigonometry.
Since angle \( A = 45^\circ \), we can use the trigonometric relationship for the height of the triangle: \[ \text{Height} = 10\sqrt{2} \times \sin(45^\circ) = 10 \text{ units}. \]

Step 4: Solving for \( BD \).
Now that we know the height of triangle \( ABD \), we can solve for the base \( BD \) using the area formula: \[ 200 = \frac{1}{2} \times BD \times 10 \] Solving this equation for \( BD \): \[ BD = 40 \text{ units}. \]

Step 5: Finding the Sum of the Parallel Sides.
The sum of the lengths of the parallel sides is the total length of the non-parallel sides, plus the base \( BD \). Therefore, the sum of the parallel sides is: \[ 40 + 40 = 60 \text{ units}. \] Final Answer: The correct answer is (a) 60.