Solving for the Radius of the Circumcircle of a Right-Angled Triangle
Introduction: This article discusses the process of determining the radius of the circumcircle of a right-angled triangle when given certain conditions. Understanding the properties of right-angled triangles, their circumcircles, and application of theorems is essential for solving such mathematical problems.
Understanding the Given Conditions
The problem provides a right-angled triangle whose area is 24 cm2 and the length of the hypotenuse is 10 cm. The goal is to find the radius of the circumcircle of this triangle.
Step-by-Step Solution:
In a right-angled triangle, the hypotenuse is always a diameter of the circumcircle. This is a key property to remember when solving such problems.
Using the Diameter Property
The radius of the circumcircle is half the length of the hypotenuse. Given that the hypotenuse is 10 cm, the radius (R) is calculated as:
Radius (R) Diameter / 2 10 cm / 2 5 cm
Alternative Approach Using Area
Alternatively, we can use the area of the triangle to solve for the radius. The area (A) of the right-angled triangle is given as 24 cm2. The formula for the area of a triangle is:
A (1/2) * base * height
For a right-angled triangle, we can write:
A (1/2) * b * h 24 cm2
Using the formula for the circumradius (R) of a right-angled triangle:
R c / 2
Where c is the hypotenuse. Substituting the given value of the hypotenuse:
R 10 cm / 2 5 cm
Publishable Content:
The problem in question is solving for the radius of the circumcircle of a right-angled triangle with an area of 24 cm2 and a hypotenuse of 10 cm.
1. Using the Diameter Property:
In a right-angled triangle, the hypotenuse is the diameter of the circumcircle. Therefore, the radius (R) is:
R 10 cm / 2 5 cm
2. Using the Area Formula:
The area (A) of the triangle is given by:
A (1/2) * b * h 24 cm2
Using the formula for the circumradius (R) of a right-angled triangle:
R c / 2 10 cm / 2 5 cm
Conclusion:
Both methods confirm that the radius of the circumcircle is 5 cm. This calculation is based on the property that the hypotenuse of a right-angled triangle is the diameter of the circumcircle.
Frequently Asked Questions (FAQs):
Q1: In a right-angled triangle, the hypotenuse is always a diameter of?
A1: The hypotenuse of a right-angled triangle is always a diameter of the circumcircle.
Q2: How do you find the circumradius of a right-angled triangle?
A2: The circumradius (R) of a right-angled triangle can be found using the formula R c / 2, where c is the length of the hypotenuse.
Q3: If the area of a right-angled triangle is 24 cm2 and the hypotenuse is 10 cm, what is the radius of the circumcircle?
A3: The radius of the circumcircle is 5 cm, as the hypotenuse (10 cm) is the diameter of the circumcircle.