<p>In a three dimensional coordinate system P, Q and R are images of a point A(a, b, c) in the XY, YZ and the ZX planes respectively. If G is the centroid of triangle PQR, then area of triangle AOG is (O is the origin)</p>
Step-by-Step Solution
Key Concept: Find the images of point A in the three coordinate planes, then determine the centroid G of triangle PQR. Use the distance formula to calculate the area of triangle AOG where O is the origin.
Step 1: Find images in coordinate planes Point A(a, b, c) • Image in XY plane: P(a, b, 0) [z-coordinate becomes 0] • Image in YZ plane: Q(0, b, c) [x-coordinate becomes 0] • Image in ZX plane: R(a, 0, c) [y-coordinate becomes 0] Step 2: Find centroid G of triangle PQR G = ((a+0+a)/3, (b+b+0)/3, (0+c+c)/3) G = (2a/3, 2b/3, 2c/3) Step 3: Find vectors OA and OG OA = (a, b, c) OG = (2a/3, 2b/3, 2c/3) Step 4: Check if O, A, G are collinear OG = (2/3)OA Since G lies on line OA (G is 2/3 of the way from O to A), the three points O, A, and G are collinear. Step 5: Calculate area of triangle AOG When three points are collinear, they do not form a proper triangle. The area is 0. ∴ Answer: Unknown
Correct Answer: Unknown