Directions: Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. What is the area of the right-angled triangle? (Bank Recruitment, 2009) I. Height of the triangle is three-fourths of the base. II. Hypotenuse of the triangle is 5 metres.

### Concept & Data Sufficiency (Geometry) To find the area of a right-angled triangle, we need both the base and the height, or enough information to determine them. The area formula is: $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ ### Step-by-Step Solution * **Analyze Statement I:** Height is three-fourths of the base. Let base = $x$, then height = $\frac{3}{4}x$. We cannot find the exact area without knowing $x$. So, I alone is not sufficient. * **Analyze Statement II:** Hypotenuse = 5 m. We cannot find the individual base and height from just the hypotenuse. So, II alone is not sufficient. * **Combine I and II:** Using the Pythagorean theorem: $\text{base}^2 + \text{height}^2 = \text{hypotenuse}^2$. * $x^2 + (\frac{3}{4}x)^2 = 5^2$ * $x^2 + \frac{9}{16}x^2 = 25$ * $\frac{25}{16}x^2 = 25 \implies x^2 = 16 \implies x = 4$. * Base = 4, Height = 3. Now we can find the area ($\frac{1}{2} \times 4 \times 3 = 6$). * Both statements together are necessary. ### Exam Strategy & Shortcut For data sufficiency, you don't need to fully solve for the final value. Once you establish that combining the equations gives exactly one unique positive solution for the variables needed, you can immediately conclude both are necessary and move on. ### Common Pitfall Assuming that knowing the ratio of sides (Statement I) is enough to find the area, forgetting that area is an absolute value requiring actual lengths. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question**.