Question

What is the perimeter of isosceles triangle \( MNP \)? (1) \( MN = 16 \) (2) \( NP = 20 \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

For isosceles triangles, remember that two sides are equal, which simplifies the process of calculating the perimeter.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that \( MN = 16 \), but we do not have enough information about the lengths of the other sides of the triangle or its base, so statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( NP = 20 \), but without knowing the length of the other side of the triangle, we cannot determine the perimeter of the triangle. Thus, statement (2) alone is also insufficient.
Step 3: Combine both statements.
Since the triangle is isosceles, the two other sides must be equal in length. We know \( MN = 16 \) from statement (1), and \( NP = 20 \) from statement (2). The perimeter of the triangle is: \[ \text{Perimeter} = 2 \times \text{equal sides} + \text{base} = 2 \times 16 + 20 = 32 + 20 = 52 \] Thus, the perimeter is 52, and both statements together provide enough information to solve for the perimeter.
\[ \boxed{52} \]