Question

If arc XYZ above is a semicircle, what is its length?
1. q = 2
2. r = 8

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

Hint:

Recognize key geometric configurations. An angle inscribed in a semicircle is always a right angle. The altitude to the hypotenuse of a right triangle creates similar triangles and leads to the geometric mean theorem (\(h^2=p_1 p_2\)), which is fundamental in solving this problem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the length of the semicircle arc XYZ. The formula for the circumference of a full circle is \(C = 2\pi R\) or \(C = \pi D\), where R is the radius and D is the diameter. The length of a semicircle arc is half the circumference, so \(L = \frac{1}{2} \pi D = \pi R\). To find the length, we need to determine the diameter (or radius) of the semicircle. The diameter of the semicircle is the segment XZ, which has a length of \(q+r\).

Step 2: Key Formula or Approach:
In the given figure, if we connect points X to Y and Z to Y, we form a triangle \(\triangle XYZ\). Since \(\triangle XYZ\) is inscribed in a semicircle with diameter XZ, the angle at Y, \(\angle XYZ\), must be a right angle (90°). The line segment of length 4 is the altitude from the right angle to the hypotenuse. There is a geometric theorem (the altitude theorem) for right triangles which states that the square of the altitude from the right angle to the hypotenuse is equal to the product of the two segments it divides the hypotenuse into.
In this case, the altitude is 4, and the segments of the hypotenuse (the diameter) are q and r. So, we have the relationship:
\[ 4^2 = q \times r \] \[ 16 = qr \] Our goal is to find the value of \(q+r\) to determine the diameter.

Step 3: Detailed Explanation:
Analyzing Statement (1): q = 2.
Using the relationship \(qr = 16\), if we substitute \(q=2\), we can find r:
\[ 2 \times r = 16 \] \[ r = 8 \] The diameter D is \(q+r\).
\[ D = 2 + 8 = 10 \] The radius R is \(D/2 = 5\).
The length of the semicircle arc is \(L = \pi R = 5\pi\).
Since we found a unique value for the length, statement (1) is sufficient.
Analyzing Statement (2): r = 8.
Using the relationship \(qr = 16\), if we substitute \(r=8\), we can find q:
\[ q \times 8 = 16 \] \[ q = 2 \] The diameter D is \(q+r\).
\[ D = 2 + 8 = 10 \] The radius R is \(D/2 = 5\).
The length of the semicircle arc is \(L = \pi R = 5\pi\).
Since we found a unique value for the length, statement (2) is also sufficient.

Step 4: Final Answer:
Both statements, independently, provide enough information to find the length of the arc. Therefore, each statement alone is sufficient.