Question

Each of the following questions is followed by two statements, I and II. Answer as follows:
Mark option (1) if the question can be answered using statement I alone, but not using statement II alone.
Mark option (2) if the question can be answered using statement II alone, but not using statement I alone.
Mark option (3) if the question can be answered using both statements I and II together, but not by using either statement alone.
Mark option (4) if the question cannot be answered even using both statements I and II together.

What is the area of the isosceles trapezium?
I. Of the two parallel sides, one is 6 cm smaller than the other.
II. The line joining the midpoints of the non-parallel sides is 13 cm in length and is at a distance of 2 cm from the base.

• A. The question can be answered using statement I alone, but not using statement II alone.
• B. The question can be answered using statement II alone, but not using statement I alone.
• C. The question can be answered using both statements I and II together, but not by using either statement alone.
• D. The question cannot be answered even using both statements I and II together.

Hint:

Recall that the segment joining the midpoints of the legs of a trapezium equals the average of the two parallel sides and lies exactly midway in height between them; check which statement gives both the sum of the parallel sides and the full height.

Answer 1

The Correct Option is B

Step 1: Picture the trapezium and place coordinates.
Put the two parallel sides horizontal: the longer one of length $a$ on the bottom, the shorter one of length $b$ on top, at height $h$ above the bottom.
Because the area only needs $(a+b)$ and $h$, that is what we should hunt for in each statement.


Step 2: Test statement I by itself.
Statement I only says $a - b = 6$.
Knowing just the difference of two numbers does not fix their sum. For instance $a=10,b=4$ or $a=20,b=14$ both satisfy $a-b=6$ but give different sums, $14$ and $34$.
There is also no height given here. So statement I alone leaves the area completely undetermined.


Step 3: Derive what the midpoint segment tells us, from scratch.
Let $E$ and $F$ be the midpoints of the two slanted, non-parallel sides. The segment $EF$ joining these midpoints always runs parallel to the two bases and has length equal to the average of the two bases: $EF = \frac{a+b}{2}$.
Also, since $E$ and $F$ are midpoints of the slanted sides, the segment $EF$ sits exactly halfway up between the bottom base and the top base, that is, at height $h/2$ from each base.


Step 4: Plug in statement II's numbers.
Statement II gives $EF = 13$, so $\frac{a+b}{2} = 13$, hence $a + b = 26$.
Statement II also gives the distance from $EF$ to the base as $2$ cm, and since $EF$ is at height $h/2$, we get $\frac{h}{2} = 2$, so $h = 4$.
Both pieces we need, $a+b = 26$ and $h = 4$, are now known using statement II only.


Step 5: Compute the area.
\[ \text{Area} = \frac{1}{2}(a+b)h = \frac{1}{2}(26)(4) = 52 \text{ cm}^2 \]
Since statement I was never used in this computation, it adds nothing extra; statement II alone was enough all along.


Final Answer:
The area is $52 \text{ cm}^2$, found using statement II alone, so option (2) is correct.
\[ \boxed{52 \text{ cm}^2 \text{, option (2)}} \]