Let A \((x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B \((1, 4, -1)\) and C \((2, 0, -2)\). Then among the statements:
(S1): ABC is an isosceles right angled triangle, and
(S2): the area of \(\triangle ABC\) is \( \frac{9\sqrt{2}}{2} \).
(S2): the area of \(\triangle ABC\) is \( \frac{9\sqrt{2}}{2} \).
Show Hint
- only (S1) is true
- both are true
- only (S2) is true
- both are false
The Correct Option is D
Solution and Explanation
Step 1: Calculate the distances. Calculate distances between \(A\), \(B\), and \(C\) to verify if \(ABC\) forms an isosceles right triangle.
Step 2: Verify statement (S1). Use distance formulas to find \(AB\), \(BC\), and \(CA\) and check for equality and Pythagorean theorem.
Step 3: Verify statement (S2). Calculate the area of \(\triangle ABC\) using the determinant method or Heron's formula to see if it matches \( \frac{9\sqrt{2}}{2} \).
Step 4: Conclusion for each statement. Determine the truth of each statement based on calculations.
Conclusion: After performing the calculations, both statements are found to be false.
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