To solve this problem, let's break it down step by step.

Given:

• Two circles are inscribed in a square.
• The circles are tangent to each other, and their centers are on the square's diagonal.
• Each circle has a radius of 1 unit.

Objective:

Find the diagonal length of the square.

Step 1: Understanding the Geometry

Since the two circles are tangent to each other, their combined diameters span part of the diagonal of the square. The radius of each circle is given as 1 unit, so their diameters are each $2 \times 1 = 2$ units.

The centers of the two circles lie along the diagonal of the square, and the circles are tangent to each other, meaning the distance between their centers is exactly the sum of their radii: $2 \times 1 = 2$ units.

Step 2: The Geometry of the Square

The diagonal of the square also passes through the centers of the two circles. Because the square is symmetrical and the circles are tangent to each other and the square’s sides, the problem essentially forms a right triangle with half of the diagonal.

Let’s say the length of the square’s side is $s$.

• The relationship between the diagonal $d$ and the side $s$ of a square is given by the Pythagorean theorem: $d = s\sqrt{2}$

Step 3: Analyzing the Arrangement

Since the circles are tangent to each other and the sides of the square, the total length of the diagonal includes:

• Two radii (one from each circle), contributing $1 + 1 = 2$ units.
• Plus, there is additional space between the two outermost points of tangency (between the circles and the square’s corners) which forms the rest of the diagonal.

Step 4: Solving for the Diagonal Length

The centers of the circles lie on the diagonal, and the distance between the two centers is 2 units. This 2-unit distance represents a portion of the diagonal. The rest of the diagonal is formed by the distances from each circle’s center to the square’s corners. Each of these distances is equal to the radius of a circle, which is 1 unit for each circle.

Thus, the total diagonal is the sum of these distances: $d = 1 + 2 + 1 = 4 \, \text{units}$

Final Answer:

The length of the diagonal of the square is 4 units.

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Do you want further details or have any questions about this solution?

Here are 5 related questions:

1. How can we derive the diagonal-length formula for a square?
2. What is the relation between a circle's radius and the area of the circle?
3. Can you solve a similar problem with different circle sizes?
4. How would the problem change if the square was rotated?
5. How does the Pythagorean theorem apply to square diagonals in general?

Tip: When dealing with inscribed figures, always try to find key geometric relations like symmetry and tangency.