Geometry with Two Circles $$ x = 25 \text{ meters} $$

Geometry with Two Circles

In this lesson, we explore a geometry problem involving two circles placed inside a rectangle. By using coordinate geometry and the distance formula, we can find the unknown length of the rectangle in a clear and systematic way.

1. Theory of Geometry with Circles

When two circles touch each other externally, the distance between their centers is equal to the sum of their radii. By placing the centers of the circles on a coordinate plane, we can use the distance formula to analyze their positions and solve geometric problems accurately.

2. Concept Explanation

In this problem, the rectangle has a fixed height, and two circles of different sizes fit inside it while touching each other. By assigning coordinates to the centers of the circles, we translate the geometric situation into an algebraic equation that can be solved step by step.

3. Steps or Formula

1. Place the centers of the circles on a coordinate plane.
2. Use the fact that touching circles have center distances equal to the sum of their radii.
3. Apply the distance formula.
4. Solve the resulting equation.
5. Add the necessary radius to find the total length of the rectangle.

4. Example Problem

Given:

A rectangle has a height of 18 meters. Inside the rectangle are two circles: a large circle with radius 8 meters and a smaller circle with radius 5 meters.

The center of the large circle is placed at:

$$ (8, 8) $$

The center of the smaller circle is placed at:

$$ (X, 13) $$

Step 1: Use the distance between centers.

Since the circles touch each other, the distance between their centers is:

$$ 8 + 5 = 13 $$

Step 2: Apply the distance formula.

$$ \sqrt{(X - 8)^2 + (13 - 8)^2} = 13 $$

Step 3: Simplify and solve.

$$ (X - 8)^2 + 5^2 = 169 $$ $$ (X - 8)^2 + 25 = 169 $$ $$ (X - 8)^2 = 144 $$ $$ X - 8 = 12 $$ $$ X = 20 $$

Step 4: Find the length of the rectangle.

The total length x is the x-coordinate of the smaller circle plus its radius:

$$ x = 20 + 5 = 25 $$

5. Real-World Application

This type of analytic geometry problem is useful in technical design, layout planning, and space optimization, where precise positioning of circular objects is required.

Final Answer:

$$ x = 25 \text{ meters} $$