Question

What is the area of the triangle?
Statement I
I. Two sides are 41 cm each.
Statement II
II. The altitude to the third side is 9 cm long.

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statement I and statement II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

When height is given, you must also know the corresponding base to find the are(a) Sides help derive the base via Pythagoras.

Answer 1

The Correct Option is C

Step 1: Analysing Statement I Knowing two equal sides (41 cm) only tells us it is isosceles but not the base or height, so area cannot be compute(d) Step 2: Analysing Statement II Knowing only the height (9 cm) to the base is not enough — we also need the length of the base. Step 3: Combining Statements From I: Two sides of 41 cm each and the altitude in II allow use of Pythagoras to find the base: Let base = $b$, then $(b/2)^2 + 9^2 = 41^2 \Rightarrow (b/2)^2 = 1681 - 81 = 1600 \Rightarrow b/2 = 40 \Rightarrow b = 80$. Area = $\frac{1}{2} \times b \times h = \frac{1}{2} \times 80 \times 9 = 360 \ \text{cm}^2$. Step 4: Conclusion Both statements are neede(d)