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 radius of smaller circle and larger circle

well, of course, the larger circle has radius 1.

Draw a diagram of the upper right quarter of the square. Include half the diagonal of the square, which goes through the centers of the two circles (O and Q), which are tangent at point P.

Draw a vertical diameter of the smaller circle, passing through its center Q, and intersects the side of the square at R.

Draw a small horizontal segment from P to where it intersects the vertical diameter through Q at S.

Adding up the vertical pieces,

OP = 1 (the radius of the larger circle)
PQ = QR = r (the radius of the smaller circle)

SQ = r/√2

1/√2 + r/√2 + r = 1
so,
r = (√2-1)/(√2+1)

A piece of wire 30 cm long is cut into two pieces. One of these is bent into a circle, and the other is bent into a square enclosing the circle, as shown in the diagram. What is the diameter of the circle? (Remember that the circumference of a circle = 2 pi r.

To determine the radius of the smaller circle, we need to consider the properties of the inscribed square.

Step 1: Find the diagonal of the square
The diagonal of a square can be found using the formula:

Diagonal = side length x √2

Since the radius of the larger circle is 1, the side length of the square is twice the radius, which is 2.

Diagonal = 2 x √2

Step 2: Find the side length of the square
The side length of the square can be found using the formula:

Side length = Diagonal / √2

Substituting the value of the diagonal from Step 1:

Side length = (2 x √2) / √2

Simplifying:

Side length = 2

Step 3: Find the radius of the smaller circle
Since the smaller circle is tangent to two sides of the square and the larger circle, the distance from the center of the smaller circle to any of the sides of the square is equal to the radius of the larger circle, which is 1.

Therefore, the radius of the smaller circle is also 1.

In conclusion, the radius of the smaller circle is 1 and the radius of the larger circle is also 1.

To determine the radius of the smaller circle, we need to look at the key feature of the problem: the tangent points.

Let's consider the tangent points of the smaller circle with the square and the larger circle.

1. Tangent with the square:
Since the square has sides equal to the diameter of the larger circle, the smaller circle will be tangent to the square at the midpoints of two sides. Let's denote this tangent point as A.

2. Tangent with the larger circle:
The smaller circle is also tangent to the larger circle. Let's denote this tangent point as B.

Now, let's draw the radius of the larger circle from its center to the tangent point B. This radius has a length equal to the sum of the radius of the smaller circle and the radius of the larger circle.

Given that the radius of the larger circle is 1, let's represent the radius of the smaller circle as r.

Using the Pythagorean theorem, we can calculate the length of this radius as follows:

(r + 1)^2 = (2r)^2

Simplifying this equation, we have:

r^2 + 2r + 1 = 4r^2

Subtracting 4r^2 from both sides:

3r^2 - 2r - 1 = 0

Now, we can solve this quadratic equation for r using the quadratic formula:

r = (-(-2) ± √((-2)^2 - 4(3)(-1))) / (2(3))

Simplifying further:

r = (2 ± √(4 + 12)) / 6

r = (2 ± √16) / 6

r = (2 ± 4) / 6

Thus, we have two possible values for r:

r₁ = (2 + 4) / 6 = 6/6 = 1

r₂ = (2 - 4) / 6 = -2/6 = -1/3

Since radius cannot be negative, the radius of the smaller circle is 1.

Hence, the radius of the smaller circle is 1, and the radius of the larger circle is also 1.