Let’s analyze and solve the problem step by step:

Problem Overview

Two circles intersect at points $C$ and $D$. The details given are:

• Radius of circle $A$: $24 \, \text{cm}$,
• Radius of circle $B$: $7 \, \text{cm}$,
• Distance between centers $AB$: $25 \, \text{cm}$,
• You are asked to: a) Prove that $\angle ACB = 90^\circ$, b) Calculate: i) The size of obtuse $\angle CBB'$, ii) The area of the shaded region.

a) Show that $\angle ACB = 90^\circ$:

Proof:

1. Consider the triangle formed by $\triangle ABC$.

   • $AC = 24 \, \text{cm}$ (radius of circle $A$),
   • $BC = 7 \, \text{cm}$ (radius of circle $B$),
   • $AB = 25 \, \text{cm}$ (distance between the centers).

2. Use the Pythagorean theorem to verify if $\triangle ABC$ is a right triangle: $AB^2 = AC^2 + BC^2$ Substitute the values: $25^2 = 24^2 + 7^2$ $625 = 576 + 49$ $625 = 625$

3. Since the Pythagorean theorem holds, $\triangle ABC$ is a right triangle, and $\angle ACB = 90^\circ$.

b) Calculations

i) Find the size of obtuse $\angle CBB'$:

The obtuse angle $\angle CBB'$ is supplementary to $\angle CBA$.

1. $\triangle ABC$ is a right triangle, so: $\angle CBA = 90^\circ$

2. The obtuse angle is: $\angle CBB' = 180^\circ - \angle CBA$ $\angle CBB' = 180^\circ - 90^\circ = 90^\circ$

Thus, $\angle CBB' = 90^\circ$.

ii) Calculate the area of the shaded part:

The shaded part is a combination of two circular segments:

1. Circular segment of circle $A$:

   • The sector subtended by $\angle ACB$ is a quarter circle, since $\angle ACB = 90^\circ$.
   • Area of sector of $A$: $\text{Area} = \frac{\theta}{360^\circ} \cdot \pi r^2$ Substitute $\theta = 90^\circ$ and $r = 24 \, \text{cm}$: $\text{Area} = \frac{90}{360} \cdot \pi (24)^2$ $\text{Area} = \frac{1}{4} \cdot \pi \cdot 576 = 144\pi \, \text{cm}^2$

2. Circular segment of circle $B$:

   • The sector subtended by $\angle CBB' = 90^\circ$ is also a quarter circle.
   • Area of sector of $B$: $\text{Area} = \frac{90}{360} \cdot \pi (7)^2$ $\text{Area} = \frac{1}{4} \cdot \pi \cdot 49 = 12.25\pi \, \text{cm}^2$

3. Shaded area (overlap of the segments): The shaded area is: \text{Shaded Area} = \text{Area of sector of A} + \text{Area of sector of B} Substitute: $\text{Shaded Area} = 144\pi + 12.25\pi = 156.25\pi \, \text{cm}^2$ Approximate using $\pi \approx 3.1416$: $\text{Shaded Area} \approx 156.25 \cdot 3.1416 = 491.07 \, \text{cm}^2$

Final Answers:

a) $\angle ACB = 90^\circ$ (proved).

b) i) $\angle CBB' = 90^\circ$.

ii) Shaded area $\approx 491.07 \, \text{cm}^2$.

Would you like detailed explanations for any specific step or further examples? Here are some related questions to explore:

1. How do you calculate the area of a segment of a circle?
2. What are the key properties of right-angled triangles?
3. How is the Pythagorean theorem applied in various geometric problems?
4. What are the practical applications of circle segments and sectors?
5. How to handle problems involving overlapping circles?

Tip: Always double-check your geometric constructions for clarity in diagrams.