ABC is a sector of a circle with radius R and center C. The arc DE lies on a circle also centred at C. If DE divides ABC into two regions of equal area, find the length of CD in terms of R
If we call DE r, then we have
r^2θ = 1/2 R^2θ
r^2 = R^2/2
r = R/√2
or, since the areas are in the ratio 1:2, the radii are in the ratio 1:√2
To solve this problem, we need to use the properties of a circle, specifically the area of a sector and the properties of equal areas.
Let's start by understanding the given information:
1. ABC is a sector of a circle with radius R and center C.
2. The arc DE lies on a circle also centered at C.
3. The areas of the two regions divided by the arc DE are equal.
Now, let's work through the solution step by step:
1. The area of a sector can be calculated using the formula:
Area of sector = (θ/360) * π * R^2,
where θ is the central angle of the sector in degrees, R is the radius of the circle.
2. The total area of the sector ABC is given as π * R^2.
3. Since the areas of the two regions divided by the arc DE are equal, each region must have an area of (1/2) * π * R^2.
4. Let's assume that the central angle of the sector ABC is θ degrees. Then the central angle of the region separated by the arc DE is (180 - θ) degrees, as the sum of the two angles is 180 degrees.
5. The area of the region separated by the arc DE can be calculated using the formula for the sector area:
Area of the region = (θ/360) * π * R^2.
Since this area is equal to (1/2) * π * R^2, we can equate the two and solve for θ:
(θ/360) * π * R^2 = (1/2) * π * R^2
Cancelling out π * R^2 from both sides, we have:
θ/360 = 1/2
Solving for θ:
θ = 180 degrees.
6. Now that we know the central angle of the sector ABC is 180 degrees, we can calculate the length of CD using the circumference of the circle formula:
Circumference = 2 * π * R.
The length of the arc DE will be half the circumference of the circle, as the central angle is 180 degrees:
Length of DE = (180/360) * 2 * π * R = π * R.
As the arc DE separates the regions of equal area, the length of the arc CD will also be π * R.
Therefore, the length of CD is π * R in terms of R.