**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 .**

**Discussion:**

Teacher: Here is a clue: rotate \( \Delta EDC \) about point E such that EC falls along EA. Can you draw the diagram after rotation?

**Student:**

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

But

Thus

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