Math Problem Statement
Solve question
Solution
Let's solve the question step by step.
Given Information:
- There are two circles.
- The smaller circle has center and radius .
- Diameters and are perpendicular to each other.
- The larger circle has center and passes through points and .
Part (i) - Show that the radius of the larger circle is .
To show this, consider the following:
- Point is the center of the smaller circle, and , as both and are on the smaller circle.
- The diameters and are perpendicular. This means that forms a right-angled triangle, with and as two sides.
- Since and lie on the larger circle, and is its center, the distance is the radius of the larger circle.
To find :
- Use Pythagoras' Theorem in triangle .
- .
Thus, the radius of the larger circle is .
Part (ii) - Find the area of the shaded region in terms of .
The shaded region is the area between the two circles. To find this:
-
Area of the larger circle:
The radius of the larger circle is .
Hence, the area is: -
Area of the smaller circle:
The radius of the smaller circle is .
Hence, the area is: -
Shaded area:
The shaded region lies outside the smaller circle and inside the larger circle, so:
Therefore, the area of the shaded region is .
Would you like more details on any step or have any questions?
Here are 5 related questions for further practice:
- How would the result change if the radius of the smaller circle were doubled?
- What is the length of the arc formed by in the larger circle?
- How would you find the circumference of both circles?
- What would happen to the area of the shaded region if the radius of the smaller circle is halved?
- How can you derive the area of the sector for the larger circle that spans angle ?
Tip:
When dealing with perpendicular diameters in a circle, using Pythagoras' Theorem is often helpful to find distances between points.
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Math Problem Analysis
Mathematical Concepts
Geometry
Circles
Pythagorean Theorem
Area Calculation
Formulas
Distance formula: BC = √(AC^2 + AB^2) = √(r^2 + r^2) = r√2
Area of a circle: A = πr²
Theorems
Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Suitable Grade Level
Grades 11-12