Math Problem Statement

Solve question

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.

Solution

Let's solve the question step by step.

Given Information:

  • There are two circles.
  • The smaller circle has center AA and radius rr.
  • Diameters CADCAD and BAEBAE are perpendicular to each other.
  • The larger circle has center BB and passes through points CC and DD.

Part (i) - Show that the radius of the larger circle is r2r\sqrt{2}.

To show this, consider the following:

  1. Point AA is the center of the smaller circle, and AC=AD=rAC = AD = r, as both CC and DD are on the smaller circle.
  2. The diameters CADCAD and BAEBAE are perpendicular. This means that ABC\triangle ABC forms a right-angled triangle, with AC=rAC = r and ABAB as two sides.
  3. Since CC and DD lie on the larger circle, and BB is its center, the distance BCBC is the radius of the larger circle.

To find BCBC:

  • Use Pythagoras' Theorem in triangle ABCABC.
  • BC=AB2+AC2=r2+r2=2r2=r2BC = \sqrt{AB^2 + AC^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2}.

Thus, the radius of the larger circle is r2r\sqrt{2}.

Part (ii) - Find the area of the shaded region in terms of rr.

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 r2r\sqrt{2}.
    Hence, the area is: Area of larger circle=π(r2)2=2πr2.\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 rr.
    Hence, the area is: Area of smaller circle=πr2.\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: Shaded area=Area of larger circleArea of smaller circle=2πr2πr2=πr2.\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 πr2\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 CDCD 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 CAD\angle CAD?

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