How to find circumradius of a triangle with help of coordinates
The Correct Answer and Explanation is:
To find the circumradius (R) of a triangle using the coordinates of its vertices, use the formula:R=abc4AR = \frac{abc}{4A}R=4Aabc
Where:
- aaa, bbb, and ccc are the lengths of the sides of the triangle
- AAA is the area of the triangle
Step-by-step Method:
Assume the vertices of the triangle are A(x1,y1)A(x_1, y_1)A(x1,y1), B(x2,y2)B(x_2, y_2)B(x2,y2), and C(x3,y3)C(x_3, y_3)C(x3,y3)
1. Calculate the lengths of the sides:
a=distance between B and C=(x3−x2)2+(y3−y2)2a = \text{distance between B and C} = \sqrt{(x_3 – x_2)^2 + (y_3 – y_2)^2}a=distance between B and C=(x3−x2)2+(y3−y2)2b=distance between A and C=(x3−x1)2+(y3−y1)2b = \text{distance between A and C} = \sqrt{(x_3 – x_1)^2 + (y_3 – y_1)^2}b=distance between A and C=(x3−x1)2+(y3−y1)2c=distance between A and B=(x2−x1)2+(y2−y1)2c = \text{distance between A and B} = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}c=distance between A and B=(x2−x1)2+(y2−y1)2
2. Find the area (A) using the determinant method:
A=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right|A=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
3. Plug the values into the circumradius formula:
R=abc4AR = \frac{abc}{4A}R=4Aabc
Explanation:
The circumradius is the radius of the circle that passes through all three vertices of a triangle. It does not depend on the triangle being right-angled, scalene, or isosceles. The formula used relates the product of the triangle’s side lengths with its area.
Using coordinates makes it possible to compute side lengths and area algebraically. Side lengths come from the distance formula, and the area comes from a determinant expression derived from the shoelace formula. The final ratio ensures the result gives the radius of the circumscribed circle. This process is especially useful in coordinate geometry and analytic geometry problems involving circles and triangles.
