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### Part (a) Prove that \( \triangle ADE \sim \triangle BCE \).

To prove that the two triangles are similar, we need to show that they have the same angles.

1. **Angle \( \angle ADE = \angle BCE \)**: 
   Both of these angles are subtended by the same arc \( \overarc{AB} \) in the circle. Since angles subtended by the same arc in a circle are equal, we have:
   \[
   \angle ADE = \angle BCE
   \]

2. **Angle \( \angle EAD = \angle ECB \)**:
   Similarly, both of these angles are subtended by arc \( \overarc{DC} \), so they are equal by the same reasoning.

3. **Angle \( \angle DEA = \angle CEB \)**: 
   These are vertically opposite angles and are therefore equal.

Since all corresponding angles are equal, by the **AA similarity criterion**, we can conclude that:
\[
\triangle ADE \sim \triangle BCE
\]

---

### Part (b) (i) Find the radius of the circle.

We are given:
- \( BC = 60 \, \text{cm} \)
- \( CE = 21 \, \text{cm} \)
- \( BE = 45 \, \text{cm} \)
- \( AD \) is the diameter, and the ratio \( AD : AB = 3:5 \).

From this, we can deduce that \( AB = \frac{5}{8} AD \).

To find the radius, we first recognize that:
- \( AD \) is the diameter of the circle, so the radius \( r = \frac{AD}{2} \).

Next, using the intersecting chords theorem (Power of a Point Theorem):
\[
AE \times ED = BE \times EC
\]
Let \( AE = x \) and \( ED = AD - x \). Then we have:
\[
x \times (AD - x) = 45 \times 21
\]
Solving this will give us the value of \( x \), and then we can use it to find \( AD \) and the radius.

Let me calculate that next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed with solving this step manually, or should we try again later? Here’s a quick recap:

We are setting up the Power of a Point theorem, which gives:
\[
AE \times ED = BE \times EC = 45 \times 21 = 945.
\]
With \( AD \) as the diameter, the relationship between \( AE \) and \( ED \) can be solved from this equation.

Let me know how you’d like to continue!

Here are some questions that may help clarify things:
1. What additional information do we have about \( AD \)?
2. How do we find the radius once \( AD \) is calculated?
3. Can we use other properties of the circle to find \( AE \)?
4. What conditions make triangles similar in circles?
5. How is the Power of a Point theorem used to find segment lengths?

**Tip**: The intersecting chords theorem is a powerful tool to calculate unknown chord lengths from given segments.

In the figure, ABCD is a circle. Chords AC and BD intersect at E. (a) Prove that △ADE ∼ △BCE. (b) It is given that AD is a diameter of the circle. Suppose that BC = 60 cm, CE = 21 cm, BE = 45 cm, and AB:AD = 3:5. (i) Find the radius of the circle. (ii) Find the perimeter of △ABE.

This circle geometry problem involves proving the similarity of triangles △ADE and △BCE, and calculating the radius of the circle and the perimeter of △ABE. Given BC = 60 cm, CE = 21 cm, BE = 45 cm, and AB:AD = 3:5, use the Power of a Point Theorem to find unknown lengths. Suitable for students in grades 10-12.

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