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: Recall the area formula and set up variables.
For a trapezium with parallel sides \(a\) and \(b\) and height \(h\), the area is
\[ \text{Area} = \frac{1}{2}(a+b)h \]
So we need to know \(a+b\) and \(h\) to find the area. We do not need \(a\) and \(b\) individually.

Step 2: Check statement I alone.
Statement I tells us one parallel side is 6 cm smaller than the other, so \( a - b = 6 \).
This gives only the difference of the two sides, not their sum, and it says nothing about the height \(h\).
We cannot compute the area from this alone. Statement I alone is not sufficient.

Step 3: Check statement II alone.
The line joining the midpoints of the two non-parallel sides is called the midsegment (or median) of the trapezium.
A key property: the midsegment length equals half the sum of the parallel sides, \( m = \frac{a+b}{2} \), and the midsegment lies exactly halfway between the two parallel sides.
Statement II gives \( m = 13 \) cm, so \( \frac{a+b}{2} = 13 \), which means \( a + b = 26 \).
It also says the midsegment is 2 cm from the base (one parallel side). Since the midsegment sits exactly midway between the two parallel sides, it is also 2 cm from the other parallel side.
So the full height between the two parallel sides is \( h = 2 + 2 = 4 \) cm.
Both quantities needed for the area formula, \(a+b = 26\) and \(h = 4\), come from statement II alone.

Step 4: Compute the area using statement II.
\[ \text{Area} = \frac{1}{2}(a+b)h = \frac{1}{2}(26)(4) = 52 \text{ cm}^2 \]
This is a fixed, unique value, so statement II alone answers the question. Statement I is not needed.

Final Answer:
Statement II alone is sufficient; statement I alone is not.
\[ \boxed{\text{Option (2), area} = 52 \text{ cm}^2} \]