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**Miscellaneous Exercise**

**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

BD =

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,

and

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.