Question

Find the area of a right angle triangle whose base is 12 inches.
1. The hypotenuse is 13 inches.
2. The perpendicular height of the triangle is one less than half its base.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

In Data Sufficiency questions, your goal is not to find the final numerical answer but to determine if you *can* find it. As soon as you determine that a unique value can be calculated from a statement, you know that statement is sufficient. You don't need to perform the final calculation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the area of a right-angled triangle, we need the lengths of its base and perpendicular height. The question provides the base and asks for the area. Therefore, our goal is to determine if the given statements can help us find the height of the triangle.
Step 2: Key Formula or Approach:
The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For a right-angled triangle, the Pythagorean theorem applies:
\[ (\text{base})^2 + (\text{height})^2 = (\text{hypotenuse})^2 \] Step 3: Detailed Explanation:
The base of the triangle is given as 12 inches. Let the height be \( h \). We need to find the value of \( h \).
Analyzing Statement (1): The hypotenuse is 13 inches.
Using the Pythagorean theorem with base = 12 and hypotenuse = 13:
\[ 12^2 + h^2 = 13^2 \] \[ 144 + h^2 = 169 \] \[ h^2 = 169 - 144 \] \[ h^2 = 25 \] \[ h = 5 \text{ inches} \] Since we have found a unique value for the height, we can calculate the area:
\[ \text{Area} = \frac{1}{2} \times 12 \times 5 = 30 \text{ square inches} \] Thus, statement (1) alone is sufficient.
Analyzing Statement (2): The perpendicular height of the triangle is one less than half its base.
The base is 12 inches.
Half of the base is \( \frac{12}{2} = 6 \) inches.
One less than half its base is \( 6 - 1 = 5 \) inches.
So, the height \( h = 5 \) inches.
Again, we have found a unique value for the height, and we can calculate the area:
\[ \text{Area} = \frac{1}{2} \times 12 \times 5 = 30 \text{ square inches} \] Thus, statement (2) alone is sufficient.
Step 4: Final Answer:
Since each statement independently provides enough information to find the area of the triangle, the correct answer is that each statement ALONE is sufficient.