Question:hard

The vertices of a triangle are (0,0),(4,0) and (3,9). The area of the circle passing through these three points is

Updated On: Jan 15, 2026
  • \(\frac{14π}{3}\)
  • \(\frac{12π}{5}\)
  • \(\frac{123π}{7}\)
  • \(\frac{205π}{9}\)
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The Correct Option is D

Solution and Explanation

The problem requires calculating the area of the circumcircle of a triangle with vertices at (0,0), (4,0), and (3,9). This involves first determining the triangle's circumradius \( R \), with the circumcircle's area then computed as \( πR^2 \).
Step 1: Determine the side lengths of the triangle:
  • \( AB = \sqrt{(4-0)^2 + (0-0)^2} = 4 \)
  • \( BC = \sqrt{(3-4)^2 + (9-0)^2} = \sqrt{82} \)
  • \( CA = \sqrt{(3-0)^2 + (9-0)^2} = \sqrt{90} \)
Step 2: Calculate the triangle's area (A) using Heron's formula:
  1. Compute the semi-perimeter: \( s = \frac{4 + \sqrt{82} + \sqrt{90}}{2} \)
  2. Apply Heron's formula: \( A = \sqrt{s(s-4)(s-\sqrt{82})(s-\sqrt{90})} \)
Step 3: Calculate the circumradius \( R \) using the formula:
\[ R = \frac{abc}{4A} \]
where \( a = 4 \), \( b = \sqrt{90} \), \( c = \sqrt{82} \), and \( A \) is the previously calculated area.
Step 4: Compute the area of the circumcircle:
\[ \text{Area} = πR^2 \]
The value \(\frac{205π}{9}\) represents the area of the circle passing through the triangle's vertices, derived from the aforementioned geometric calculations.
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