+32 Two circles are drawn, with the same center. A chord of the large circle is drawn, so that it is tangent to the small circle. If the chord has a length of 10, then find the area of the ring-shaped region that is inside the large circle but outside the small circle.
+682 Analyzing the Problem
Understanding the Geometry:
We have two concentric circles (circles with the same center).
A chord of the larger circle is tangent to the smaller circle. This means the chord is perpendicular to the radius of the smaller circle at the point of tangency.
We need to find the area of the ring-shaped region, which is the difference in areas between the larger and smaller circles.
Key Points:
The chord of the larger circle is the diameter of the smaller circle.
The radius of the smaller circle is half the length of the chord.
Solution
Find the radius of the smaller circle:
Since the chord is the diameter, the radius of the smaller circle is 10/2 = 5 units.
Find the area of the smaller circle:
Area of a circle = πr², where r is the radius.
Area of the smaller circle = π(5)² = 25π square units.
Find the radius of the larger circle:
The radius of the larger circle is the sum of the radius of the smaller circle and the distance from the center to the chord (which is the same as the radius of the smaller circle).
Radius of the larger circle = 5 + 5 = 10 units.
Find the area of the larger circle:
Area of the larger circle = π(10)² = 100π square units.
Find the area of the ring-shaped region:
Area of the ring-shaped region = Area of the larger circle - Area of the smaller circle.
Area of the ring-shaped region = 100π - 25π = 75π square units.
Therefore, the area of the ring-shaped region is 75π square units.
