Question

1. in the circle below, the sum of which arcs is equivalent to arc ad?

arc ad = arc af + arc gd
arc ad = arc ac + arc bd
arc ad = arc bc + arc cd
arc ad = arc ab + arc bd

Explanation:

To determine which arcs sum to arc \( AD \), we analyze the positions of the points on the circle. The arc \( AD \) can be broken into two adjacent arcs that connect \( A \) to \( D \) through an intermediate point. Looking at the options:

• Option 1: \( \text{arc } AF + \text{arc } GD \) – These arcs do not connect \( A \) to \( D \) directly through adjacent arcs.
• Option 2: \( \text{arc } AC + \text{arc } BD \) – These arcs overlap or do not form a continuous path from \( A \) to \( D \).
• Option 3: \( \text{arc } BC + \text{arc } CD \) – These arcs connect \( B \) to \( C \) to \( D \), not \( A \) to \( D \).
• Option 4: \( \text{arc } AB + \text{arc } BD \) – Arc \( AB \) goes from \( A \) to \( B \), and arc \( BD \) goes from \( B \) to \( D \). Together, they form a continuous path from \( A \) to \( D \), so their sum is equivalent to arc \( AD \).

Answer:

D. \( \text{arc } AD = \text{arc } AB + \text{arc } BD \)