Question

D, E, F are the mid points of the sides AB, BC and CA of triangle ABC respectively. What is the area of DEF in square centimeters?
A. AD = 1 cm, DF = 1 cm and perimeter of DEF = 3 cm
B. Perimeter of ABC = 6 cm, AB = 2 cm, and AC = 2 cm.

• A. The questions can be answered by one of the statements, but not by the other
• B. The question can be answered using either of the two statements alone.
• C. The questions can be answered using both the statements together, but cannot be answered using either statement alone.
• D. The question cannot be answered even by using both the statements together.

Hint:

The triangle formed by joining midpoints has sides equal to half of the sides of the big triangle. See whether each statement fixes all three sides.

Answer 1

The Correct Option is B

Step 1: Recall the midpoint triangle property.
D, E, F are midpoints, so each side of DEF is parallel to a side of ABC and equals half of it: \(EF = \frac{AB}{2}\), \(DF = \frac{BC}{2}\), \(DE = \frac{AC}{2}\).
So if we know the three sides of ABC, we know the three sides of DEF and can find the area. We need all three sides of either triangle.

Step 2: Check statement A.
D is the midpoint of AB, so \(AB = 2 \times AD = 2\) cm.
DF joins the midpoints of AB and AC, so \(DF = \frac{BC}{2}\). Hence \(BC = 2\) cm.
The perimeter of DEF is 3 cm, so the perimeter of ABC is twice that, 6 cm. So \(AC = 6 - 2 - 2 = 2\) cm.
ABC has all sides 2 cm, so it is equilateral. Its area is \(\frac{\sqrt3}{4} \times 2^2 = \sqrt3\) square cm, and the area of DEF is one fourth of it, \(\frac{\sqrt3}{4}\) square cm.
So A alone is sufficient.

Step 3: Check statement B.
Perimeter 6 with \(AB = 2\) and \(AC = 2\) gives \(BC = 6 - 2 - 2 = 2\) cm.
Again ABC is equilateral with side 2 cm, so DEF has area \(\frac{\sqrt3}{4}\) square cm. B alone is also sufficient.

Step 4: Match with the four choices.
Choice 1 is wrong since both statements work.
Choice 3 is wrong since neither needs the other.
Choice 4 is wrong since the area can be found.

Final Answer:
Each statement alone gives the area \(\frac{\sqrt3}{4}\) square cm, so the correct choice is option 2. \[ \boxed{2} \]