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1. Three vertices of a parallelogram ABCD are A B and C Find the coordinates of the fourth vertex.
Ans. Let D be the fourth vertex of parallelogram ABCD.
Since, the diagonals of a parallelogram bisect each other. Therefore, the mid-point of AAAC and BD coincide.
Coordinates of mid-point of AC
Also Coordinates of mid-point of
Therefore, the coordinates of point D are
2. Find the length of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).
Ans. Given: A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0) are vertices of
Let D, E and F be the mid-points of BC, AC and AB respectively. Then
Coordinates of D = =
And AD = units
Again, Coordinates of E = =
And BE = units
Also Coordinates of F = =
And CF = units
3. If the origin is the centroid of the triangle PQR with vertices P Q and R then find the values of and
Ans. Given: P Q and R are the vertices of triangle PQR.
Coordinates of centroid of =
According to question,
Therefore, , ,
4. Find the coordinates of a point on axis which are at a distance of from the point P
Ans. Let Q be any point on axis. Then according to question,
PQ = =
Squaring both sides,
Therefore, the coordinates of point Q are (0, 2, 0) and
5. A point R with coordinate 4 lies on the line segment joining the points P and Q (8, 0, 10). Find the coordinates of the point R.
Ans. Let R be any point which divides the line segment joining P and Q in the ratio internally.
Coordinates of R =
But according to question,
Therefore, coordinates of R is
6. If A and B be the points (3, 4, 5) and respectively. Find the equation of the set of points P such that where is a constant.
Ans. Let P be any point.