https://file.aiquickdraw.com/mathbot/file/1727506277213fdays2pl.jpg

Let's solve the question step by step.

### Given Information:
- There are two circles.
- The smaller circle has center \( A \) and radius \( r \).
- Diameters \( CAD \) and \( BAE \) are perpendicular to each other.
- The larger circle has center \( B \) and passes through points \( C \) and \( D \).

### Part (i) - Show that the radius of the larger circle is \( r\sqrt{2} \).

To show this, consider the following:

1. Point \( A \) is the center of the smaller circle, and \( AC = AD = r \), as both \( C \) and \( D \) are on the smaller circle.
2. The diameters \( CAD \) and \( BAE \) are perpendicular. This means that \( \triangle ABC \) forms a right-angled triangle, with \( AC = r \) and \( AB \) as two sides.
3. Since \( C \) and \( D \) lie on the larger circle, and \( B \) is its center, the distance \( BC \) is the radius of the larger circle.

To find \( BC \):
- Use Pythagoras' Theorem in triangle \( ABC \).
- \( BC = \sqrt{AB^2 + AC^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \).

Thus, the radius of the larger circle is \( r\sqrt{2} \).

### Part (ii) - Find the area of the shaded region in terms of \( r \).

The shaded region is the area between the two circles. To find this:

1. **Area of the larger circle:**  
   The radius of the larger circle is \( r\sqrt{2} \).  
   Hence, the area is:
   \[
   \text{Area of larger circle} = \pi (r\sqrt{2})^2 = 2\pi r^2.
   \]

2. **Area of the smaller circle:**  
   The radius of the smaller circle is \( r \).  
   Hence, the area is:
   \[
   \text{Area of smaller circle} = \pi r^2.
   \]

3. **Shaded area:**  
   The shaded region lies outside the smaller circle and inside the larger circle, so:
   \[
   \text{Shaded area} = \text{Area of larger circle} - \text{Area of smaller circle} = 2\pi r^2 - \pi r^2 = \pi r^2.
   \]

Therefore, the area of the shaded region is \( \pi r^2 \).

---

Would you like more details on any step or have any questions? 

Here are 5 related questions for further practice:
1. How would the result change if the radius of the smaller circle were doubled?
2. What is the length of the arc formed by \( CD \) in the larger circle?
3. How would you find the circumference of both circles?
4. What would happen to the area of the shaded region if the radius of the smaller circle is halved?
5. How can you derive the area of the sector for the larger circle that spans angle \( \angle CAD \)?

### Tip:
When dealing with perpendicular diameters in a circle, using Pythagoras' Theorem is often helpful to find distances between points.

The diagram shows a circle with center A and radius r. Diameters CAD and BAE are perpendicular to each other. A larger circle has center B and passes through C and D. 
(i) Show that the radius of the larger circle is r√2. 
(ii) Find the area of the shaded region in terms of r.

Solve a geometry problem involving a smaller circle and a larger circle with radius r and r√2, respectively. Learn how to find the radius of the larger circle and calculate the area of the shaded region using the Pythagorean Theorem and circle area formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation