Question

What is the area of the shaded region consisting of \(\triangle ABE\) and \(\triangle ECD\)?

Statement (I): The area of the rectangle ABCD is 54 sq. units.
Statement (II): \(AE = 2 ED\).

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

In Data Sufficiency problems, if an absolute value (like the area) is requested, check if the combined statements allow you to solve for the necessary components of the geometric formula, even if individual side lengths remain unknown.

Answer 1

The Correct Option is C

Concept: The area of a triangle is calculated using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In a rectangle $ABCD$, if point $E$ lies on side $AD$, the height of both $\triangle ABE$ and $\triangle ECD$ corresponds to the breadth of the rectangle, which we denote as $AB$ (given $AB = CD$).

Step 1: Analyze the given statements for dimensions and relationships. Statement (I) provides the total area of the rectangle $ABCD$: \[ \text{Area}(ABCD) = AB \times AD = 54 \] Statement (II) provides the geometric relationship between the segments $AE$ and $ED$: \[ AE = 2 ED \]

Step 2: Combine the information to express the total side length $AD$. Since the point $E$ lies on $AD$, the total length is the sum of the two segments: \[ AD = AE + ED \] Substituting the ratio from Statement (II) into this equation: \[ AD = 2 ED + ED = 3 ED \]

Step 3: Calculate the value of the product of $AB$ and $ED$ using the area from Statement (I). Substitute the expression for $AD$ into the rectangle area formula: \[ AB \times (3 ED) = 54 \] Dividing both sides by 3 to isolate the product: \[ AB \times ED = 18 \]

Step 4: Determine the total area of the shaded region. The shaded area is the sum of the areas of the two triangles: \[ \text{Area}_{\text{shaded}} = \text{Area}(\triangle ABE) + \text{Area}(\triangle ECD) \] \[ \text{Area}_{\text{shaded}} = \left( \frac{1}{2} \times AB \times AE \right) + \left( \frac{1}{2} \times CD \times ED \right) \] Substitute $CD = AB$ and $AE = 2 ED$ into the equation: \[ \text{Area}_{\text{shaded}} = \left( \frac{1}{2} \times AB \times 2 ED \right) + \left( \frac{1}{2} \times AB \times ED \right) \] Simplify the expression: \[ \text{Area}_{\text{shaded}} = (AB \times ED) + \left( \frac{1}{2} \times AB \times ED \right) \] Substitute $AB \times ED = 18$ into the result: \[ \text{Area}_{\text{shaded}} = 18 + 9 = 27 \]