Calculating the Area of a Right Trapezium with an Incircle
Let's explore how to calculate the area of a right trapezium, given that the non-parallel long side is equal to 10 and it has an incircle with a radius of 2. This problem requires us to delve into the properties of trapeziums and the relationship between the incircle and the trapezium's dimensions.
Introduction to the Right Trapezium
A right trapezium, also known as a right trapezoid in some regions, has one right angle (90 degrees) and two parallel sides. In this case, the trapezium has a non-parallel long side (let's call it b) equal to 10 and an incircle (a circle that touches all four sides of the trapezium) with a radius (r) of 2. The incircle is a key to solving this problem because it simplifies the relationship between the sides and the diagonals.
The Role of the Incircle in the Right Trapezium
The incircle touches all four sides of the trapezium, meaning that the distance from the center of the circle to each side is equal to the radius, which is 2 in this case.
Key Formulas and Properties
The area (A) of a trapezium can be calculated using the formula: A ((a b) * h) / 2, where a and b are the lengths of the two parallel sides, and h is the height. The relationship between the sides, the height, and the incircle can help us find the unknown dimensions of the trapezium. The Pythagorean theorem can be used to solve for unknown sides if the height and radius are known.Solving for the Dimensions of the Right Trapezium
Given that the non-parallel long side (b) is 10 and the radius of the incircle (r) is 2, we can start by identifying the height (h) of the trapezium. Since the height is the diameter of the incircle for the vertical sides, h 4 (2 times the radius).
Finding the Parallel Sides
The base (a) of the right trapezium is the distance along the longer non-parallel side, and the shorter parallel side (c) can be calculated using the Pythagorean theorem. The right angle and the tangents from the circle to the sides of the trapezium help in this calculation.
Using the Pythagorean Theorem
Let's denote the shorter parallel side as c. The distance between the points of tangency on the non-parallel long side (10) is the sum of the two tangents from the circle to the base (a) and the height (h). Therefore, we have:
c (10 - c) 10
The tangents from a point outside a circle to the points of tangency are equal. So, let the tangents from the points of tangency to the base (a) be x and y, such that x y 10 and x y (a - c).
Final Area Calculation
Given that the base (a) is 2x (since the radius is 2 and the height is 4, the base of the trapezium is twice the radius, so a 4 2 6), and the shorter parallel side (c) can be calculated by the steps above, we can now find the area of the trapezium.
The area (A) is calculated as:
A ((a b) * h) / 2
Substituting the values:
A ((6 10) * 4) / 2
A (16 * 4) / 2
A 64 / 2
A 32
Conclusion
The area of the right trapezium with the non-parallel long side equal to 10 and an incircle of radius 2 is 32 square units. This solution involves understanding the properties of the trapezium and the role of the incircle in determining the dimensions and area.