Geometric Analysis: Solving for the Area of Trapezium ABCD

Introduction

In this article, we will explore a problem that involves the intersection of geometric shapes and the application of basic principles of geometry to solve for the area of a specific trapezium. We start with two circles and their tangents with known dimensions and slopes, leading us to uncover the area of the trapezium formed by these tangents and the x- and y-axes.

Given Circles and Their Properties

The problem provides us with two circles, both centered at the origin (0,0). Circle 1 has a radius of 20 cm, and Circle 2 has a radius of 40 cm. Both circles feature tangents with a slope of 45 degrees. However, these tangents occupy distinct quadrants. Circle 1's tangent, labeled AB, is in the second quadrant, while Circle 2's tangent, DC, is in the fourth quadrant. The points A and C lie on the x-axis, while B and D are on the y-axis. Our goal is to determine the area of trapezium ABCD.

Understanding the Tangent Relationships

To begin, it is crucial to understand the nature of the tangents. A tangent to a circle forms a right angle with the radius at the point of tangency. Since the tangents have a slope of 45 degrees, they are parallel to the lines y x and y -x. This implies that the tangents will intersect the axes at points that form a right angle with the radii to the points of tangency.

Calculating Coordinates of Key Points

Let's start by determining the coordinates of points A, B, C, and D.

- For Circle 1, the tangents AB and AC are at an angle 45 degrees with the x-axis. Since the radius of Circle 1 is 20 cm, we can use trigonometry to find the coordinates at these points. The tangent line AB in the second quadrant will have the equation: y -x 20 (since the line passes through the point (0, 20) with a 45-degree angle).

- Similarly, for Circle 2, the tangents DC and DL are at an angle 45 degrees with the x-axis. The radius of Circle 2 is 40 cm, so the tangent line DC in the fourth quadrant will have the equation: y x - 40 (since the line passes through the point (0, -40) with a 45-degree angle).

- The coordinates for point A can be found where the line y -x 20 intersects the x-axis (y 0). Solving for x gives us A(20, 0).

- For point C, we can find the intersection of the line y x - 40 with the x-axis (y 0), which gives us C(40, 0).

- The coordinates for point B can be found where the line y -x 20 intersects the y-axis (x 0). Solving for y gives us B(0, 20).

- For point D, we can find the intersection of the line y x - 40 with the y-axis (x 0). Solving for y gives us D(0, -40).

Area of Trapezium ABCD

Now that we have the coordinates of the points A(20, 0), B(0, 20), C(40, 0), and D(0, -40), we can proceed to calculate the area of trapezium ABCD.

The formula for the area of a trapezium is given by: Area (frac{1}{2} times (Base_1 Base_2) times Height).

In our case, the bases are the lengths of AB and CD (which are parallel and have endpoints on the y-axis), and the height is the distance between the x-axis and the y-axis, which is the x-coordinate difference between B and A or D and C.

The length of AB can be calculated as the distance between (0, 20) and (20, 0), which is (sqrt{(20-0)^2 (0-20)^2} sqrt{400 400} sqrt{800} 20sqrt{2}).

The length of CD can be calculated as the distance between (0, -40) and (40, 0), which is (sqrt{(40-0)^2 (0 40)^2} sqrt{1600 1600} sqrt{3200} 40sqrt{2}).

The height of the trapezium ABCD is the difference in the x-coordinates of B and A or D and C, which is 20 units.

Substituting the values into the area formula, we get:

(Area frac{1}{2} times (20sqrt{2} 40sqrt{2}) times 20 frac{1}{2} times 60sqrt{2} times 20 600sqrt{2}).

Simplifying, we find that the area of trapezium ABCD is 3600 cm2.

Conclusion

The trapezium ABCD has an area of 3600 cm2. This detailed geometric analysis demonstrates the application of trigonometry and the properties of circles and tangents to solve real-world problems. Understanding these principles is crucial for various fields that require spatial and numerical reasoning, such as engineering, physics, and architecture.

Keywords

trapezium geometry circles area calculation tangent lines