In the semicircle PABQ with centre O and diameter PQ, points A and B lie on the semicircle such that the central angle AOB is 64°. The chords BP and AQ intersect at point X inside the semicircle. What is the measure, in degrees, of angle AXP formed at X between segments XA and XP?

Introduction / Context:
This is a geometry problem involving a semicircle, central angles, chords and their intersection inside the circle. It tests knowledge of angle properties in circles, including relationships between central angles, inscribed angles and angles formed by intersecting chords. Such problems are common in higher level aptitude and Olympiad style examinations and require careful visualization of the figure.

Given Data / Assumptions:

• PABQ is a semicircle with centre O and diameter PQ.

• Points A and B lie on the semicircle such that central angle AOB = 64°.

• Chords BP and AQ are drawn and intersect at point X inside the semicircle.

• We need the size of angle AXP in degrees.

• All points lie on or inside the semicircle as described.

Concept / Approach:
Key circle theorems are involved. First, note that PQ is a diameter, so any angle subtending PQ at the semicircle is a right angle. Intersecting chords theorem states that the measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical opposite angle. By expressing the arcs in terms of the given central angle and the semicircle properties, we can find the required angle AXP.

Step-by-Step Solution:
Let us denote the arcs on the semicircle. The semicircle PABQ spans 180° of central angle. Given central angle AOB = 64°, so the arc AB measures 64°. The remaining arc from P to Q, excluding A and B, totals 180° − 64° = 116° and is distributed as arcs PA and BQ. Chord BP subtends some arc APQ and chord AQ subtends some arc PBQ, but we focus on arcs intercepted by angle AXP and its vertical opposite. Angle AXP is formed by chords XA and XP, coming from intersections of AQ and BP. By intersecting chords theorem, measure of angle AXP = 1/2 (measure of arc AP + measure of arc BQ). Similarly, angle AXQ would involve the complementary pair of arcs, including arc AB. Using geometric analysis (for example, analytic geometry with P at 180°, Q at 0°, and placing A and B such that central angle AOB = 64°) gives angle AXP equal to 58°. Thus, angle AXP = 58°.

Verification / Alternative check:
One way to verify this result is to place the semicircle on a coordinate plane with centre at the origin, PQ as a horizontal diameter from (−1, 0) to (1, 0) and choose A and B on the semicircle such that the central angle AOB is 64°. By computing the coordinates of P, Q, A and B, forming the equations of lines BP and AQ, finding their intersection point X and then evaluating the angle between vectors XA and XP using the dot product formula, one obtains an angle of approximately 58 degrees. This numerical verification supports the theoretical reasoning and matches the closest option.

Why Other Options Are Wrong:
The values 32°, 36° and 54° do not satisfy the angle relations that arise from the intersecting chords and central angle of 64°. They also fail when tested in a coordinate geometry construction, where the computed angle is very close to 58°. Hence these other options are inconsistent with the geometry of the diagram, while 58° aligns with both theoretical and computational approaches.

Common Pitfalls:
A frequent mistake is to assume that angle AXP is simply half of 64° or some direct combination like 90° − 32°, without carefully using circle theorems. Another error is to ignore that PQ is a diameter and that right angles appear in certain inscribed angles subtending PQ. Many students also mis apply the intersecting chords theorem by using the wrong arcs. Drawing a neat diagram and clearly marking arcs and angles is essential for avoiding such errors.

Final Answer:
The measure of angle AXP is 58°.