In the figure below, E is the midpoint of the arc ABEC and the segment ED is perpendicular to the chord BC at D. If the length of the chord AB is , and that of the segment BD is , determine the length of DC in terms of .
Teacher: Here is a clue: rotate about point E such that EC falls along EA. Can you draw the diagram after rotation?
Obviously EC will fit into EA as E is the midpoint of larger arc AC. Suppose D falls on D’. Then ED’A is the rotated form of EDC.
Teacher: Can you show that D’A and BA are the same line?
Student: Okay I can try. If we can show that , then we have shown that D’A and BA is the same line. Now (due to rotation) . But as they are the angle subtended by the same segment BE.
So we have . Therefore D’A falls on BA.
Teacher: Let’s revise the diagram then
Now can you finish the problem?
Student: Since due to rotation, we have