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 △ABC is 292.
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Step-by-Step Solution
Step 1: Find the coordinates of point A
Since point A lies in the xy -plane, its z -coordinate is 0. Let A be (x, y, 0). We are given that A is equidistant from three points: P1(0,3,2), P2(2,0,3), and P3(0,0,1). We use the distance formula to set up equations where AP1=AP2=AP3.
Step 2: Solve for x and y
We equate the squared distances to eliminate the square roots. First, we equate AP12=AP22 to get a relationship between x and y. Then, we equate AP12=AP32 to solve for y. Finally, we substitute the value of y back into the first relationship to find x. This gives us the coordinates of point A.
Step 3: Calculate side lengths of triangle ABC
Now that we have the coordinates of A, B, and C, we can calculate the lengths of the sides of △ABC using the distance formula. We find the lengths of AB, BC, and AC.
Step 4: Verify statement (S1): Isosceles right-angled triangle
We observe that AB=AC=3, which means △ABC is an isosceles triangle. To check if it's a right-angled triangle, we apply the Pythagorean theorem. Since AB2+AC2=BC2, the triangle is right-angled at A. Therefore, statement (S1) is true.
Step 5: Verify statement (S2): Area of triangle ABC
Since △ABC is a right-angled triangle with the right angle at A, we can use AB and AC as the base and height. The area of a right-angled triangle is half the product of its perpendicular sides. We calculate the area and compare it with the given value in statement (S2). Since 29=292, statement (S2) is false.