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To solve for the lengths \( AB \) and \( BC \) in the given shapes (a, b, and c), we'll apply different geometric principles for each.

### a) Rhombus:
- In the first figure, we have a rhombus with diagonal \( AC = 20 \) cm and an angle \( \angle BAC = 25^\circ \). 
- Since diagonals bisect the angles in a rhombus and also bisect each other at right angles, we can use trigonometry to find \( AB \).

1. The angle at \( \angle BAC = 25^\circ \) can help us calculate the half-diagonal.
2. Use the cosine rule or simple trigonometry.

\[
AB = \frac{AC}{2\cos(25^\circ)} = \frac{20}{2\cos(25^\circ)}
\]

Let me calculate the value of \( AB \).

### b) Circle with a chord:
- In the second figure, we have a circle with a chord \( AB \), a radius of 5 m, and an angle at the center \( \angle ACB = 70^\circ \).
- We can use the formula for the length of a chord:
  
\[
AB = 2r \sin\left(\frac{\theta}{2}\right)
\]
where \( r = 5 \) m and \( \theta = 70^\circ \).

Let me compute \( AB \) using this formula.

### c) Isosceles triangle:
- In the third figure, we have an isosceles triangle with base \( AC = 10 \) cm and angle \( \angle ACB = 40^\circ \).
- Since it is isosceles, the two sides \( AB \) and \( BC \) are equal, and we can use the sine rule or trigonometry to find \( AB \) and \( BC \).
  
The height divides the triangle into two right triangles. I'll use trigonometry (sine or cosine) to find these lengths.

Now, I'll proceed with the calculations.Here are the calculated lengths for \( AB \) in each part:

- **a) Rhombus**: \( AB \approx 11.03 \, \text{cm} \)
- **b) Circle**: \( AB \approx 5.74 \, \text{m} \)
- **c) Isosceles Triangle**: \( AB = BC \approx 5.32 \, \text{cm} \)

Let me know if you would like more detailed steps or explanations!

Here are five related questions to expand on this:

1. How would the solution change if the angle \( \angle BAC \) in the rhombus was larger?
2. Can you find the area of the rhombus given its diagonal lengths?
3. How would you compute the arc length corresponding to the chord \( AB \) in the circle?
4. What if the radius of the circle in part b) was larger? How would it affect the chord length?
5. How do we calculate the perimeter of the isosceles triangle in part c)?

**Tip**: Always remember that using trigonometric identities like sine, cosine, and tangent can simplify geometry problems involving angles and sides!

Find the length of AB and BC in the given shapes, rounding the answer to 2 decimal places.

Explore how to calculate the lengths of AB and BC in various geometric shapes, including a rhombus, circle, and isosceles triangle, using trigonometric formulas and geometric theorems. The detailed solutions include cosine rule, chord length formula, and properties of triangles. Ideal for students in Grades 9-12.

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