Points A, B, C, D, E, F, G and H lie on a circle in that order and are equally spaced, so they form the vertices of a regular octagon. What is the measure, in degrees, of angle FDH (that is, angle formed at D by chords DF and DH)?

Introduction / Context:
This question is based on a regular octagon inscribed in a circle. Eight points are evenly spaced on the circle, so consecutive points are separated by equal arcs. The task is to find a specific inscribed angle, angle FDH, which is formed at vertex D by chords DF and DH. Solving it requires understanding the relation between inscribed angles and intercepted arcs in a circle.

Given Data / Assumptions:

• Eight points A, B, C, D, E, F, G, H lie on a circle and are equally spaced.
• The points are in the order A, B, C, D, E, F, G, H around the circle.
• These points form a regular octagon.
• We must find angle FDH, with vertex at D and arms along chords DF and DH.
• The circle is regular, so the central angle for each adjacent pair of vertices is equal.

Concept / Approach:
In a regular octagon, the full circle of 360 degrees is divided into eight equal central angles, so each central angle measures 360 / 8 = 45 degrees. The inscribed angle theorem states that an inscribed angle is half the measure of the intercepted arc that does not include the vertex. To find angle FDH, we identify the arc FGH that this angle intercepts and use its measure to compute the inscribed angle at D.

Step-by-Step Solution:
Step 1: Since there are 8 equally spaced points, each adjacent arc (for example AB, BC, CD, etc.) corresponds to 360 / 8 = 45 degrees of arc. Step 2: Number the points in order around the circle: A (position 1), B (2), C (3), D (4), E (5), F (6), G (7), H (8). Step 3: Angle FDH has vertex at D and rays passing through F and H, so the inscribed angle at D intercepts the arc from F to H that does not contain D. Step 4: The arc from F (position 6) to H (position 8), avoiding D, passes through G only, so it consists of two adjacent arcs FG and GH. Step 5: Each adjacent arc is 45 degrees, so arc FG = 45 degrees and arc GH = 45 degrees. Step 6: Therefore, the intercepted arc FGH has measure 45 + 45 = 90 degrees. Step 7: By the inscribed angle theorem, angle FDH = (1 / 2) * measure of arc FGH = 90 / 2 = 45 degrees.

Verification / Alternative check:
We can also visualise the points on the circle by assigning central angles. Let D correspond to central angle 135 degrees, F to 225 degrees and H to 315 degrees. The chord DH is a diameter because the central angle between D and H is 180 degrees, and F lies midway between them in terms of arc steps. The triangle DFH has vertices on the circle, and angle FDH intercepts a 90 degree arc, which again confirms that angle FDH is 45 degrees by the inscribed angle rule.

Why Other Options Are Wrong:
22.5 degrees would correspond to half of a single 45 degree arc, not two arcs. Values such as 30 and 42.5 degrees have no natural connection to multiples of 45 degrees in a regular octagon and result from miscounting the number of arcs or misapplying the inscribed angle theorem. 60 degrees would require an intercepted arc of 120 degrees, which is not the case between points F and H when D is the vertex.

Common Pitfalls:
Students sometimes choose the wrong intercepted arc, including point D within the arc or counting the longer arc that passes almost all the way around the circle. Another common mistake is to forget that an inscribed angle is always half of the intercepted arc, not equal to it. Drawing a clear sketch, marking the positions of F, D and H, and carefully identifying the shorter arc FGH that does not contain D are reliable ways to avoid confusion.

Final Answer:
The measure of angle FDH is 45 degrees.