ABC is a sector of a circle with
radius R cm and centered at C. The arc DE lies on a circle also centered at C.If
the arc DE divides the sector ABC into two
regions of equal area.
Find the length of the interval CD in terms of R.
the area is proportional to the square of the radius
CD^2 = R^2 / 2
CD = R√2 / 2
To find the length of the interval CD in terms of R, we can start by drawing a diagram to visualize the given information.
Let's assume that the angle of the sector ABC is θ degrees. We know that the area of a sector of a circle is given by (θ/360) * π * R², where R is the radius of the circle.
Since the arc DE divides the sector ABC into two regions of equal area, we can set up an equation to solve for the length of the interval CD.
The area of the sector ABC can be divided into two equal parts by the arc DE. So, we have:
(θ/360) * π * R² = (180/360) * π * R²
Simplifying this equation, we get:
θ * R² = 180 * R²
Now, we can cancel out the R² term from both sides:
θ = 180
This means that the angle of the sector ABC is 180 degrees.
Now, CD is the radius of the circle centered at C. Since C is the midpoint of DE, CD is half the length of DE. Therefore, the length of the interval CD is equal to the radius R of the circle.