a circle of radius 1 is inscribed in a square. A smaller circle is tangent to two sides of the square and the first circle. Determine the circumradius and inradius of the smaller circle
a circle has only a radius, not an inradius and a circumradius.
To determine the circumradius and inradius of the smaller circle, we can use the properties of the circles' tangency and the relationship with the square.
Let's start by visualizing the problem:
1. Draw a square with the side length of 2 units, which will be our reference square.
The radius of the larger circle, which is inscribed in this square, is given as 1 unit.
2. Draw a smaller circle inside the larger circle, such that it is tangent to the larger circle at one point and tangent to two sides of the square at other points.
We want to find the circumradius and inradius of this smaller circle.
Now, let's solve for the circumradius of the smaller circle:
The circumradius of a circle is the distance from the center of the circle to any point on the circle itself. Since the smaller circle is tangent to the larger circle, if we draw a line connecting the centers of these two circles (which is the radius of the larger circle), it will also be the radius of the smaller circle.
Therefore, the circumradius of the smaller circle is also 1 unit.
Next, let's solve for the inradius of the smaller circle:
The inradius of a circle is the distance from the center of the circle to any of its tangent points with the sides of the square.
To find the inradius, we can consider the right triangle formed with the radius of the smaller circle as the hypotenuse, and two sides of the square as the legs of the triangle.
Since the side length of the square is 2 units, each leg of the triangle will have a length of 1 unit (half the side length of the square).
Using the Pythagorean theorem, we can calculate the inradius:
(inradius)^2 = (hypotenuse)^2 - (leg)^2
(inradius)^2 = (1)^2 - (1/2)^2
(inradius)^2 = 1 - 1/4
(inradius)^2 = 3/4
Taking the square root of both sides, we get:
inradius = sqrt(3/4) = sqrt(3)/2
Therefore, the inradius of the smaller circle is sqrt(3)/2 units.
To summarize:
- The circumradius of the smaller circle is 1 unit.
- The inradius of the smaller circle is sqrt(3)/2 units.