Question 1: The vertices of the triangle are The internal bisector of angle meets at Find the coordinates of and the length

Answer:

is the internal bisector of

Therefore divides in the ratio of

 

Question 2: A point with z-coordinate lies on the line segment joining the points Find its coordinates.

Answer:

Let divides in the ratio of

The z coordinate of is

 

Question 3: Show that the three points are collinear and find the ratio in which divides

Answer:

Let divides in the ratio of

The coordinates of is

Therefore

Hence divides in the ratio ( externally)

 

Question 4: Find the ratio in which the line joining is divided by the yz-plane.

Answer:

Given and

Let yz-plane divide at point in the ratio

Therefore we have

Since lies on yz-plane, the x-coordinate of will be zero.

Hence the yz-plane divides externally in the ratio of

 

Question 5: Find the ratio in which the line segment joining the points is divided by the plane

Answer:

Suppose the plane divides the line joining and at a point in the ratio 

Therefore the coordinates of would be:

Since lies on the plane , the coordinates of must satisfy the equation of the plane.

Hence the required ratio is ( externally)

 

Question 6: If the points are collinear, find the ratio in which divides

Answer:

Given points

Let divide at point in the ratio

Therefore we have

Given coordinates of are

Hence, divides in the ratio

 

Question 7: The mid-points of the sides of a triangle are given by Find the coordinates of

Answer:

Let   and   be the vertices of the triangle.

And let be the midpoint of the slides and respectively.

Now, is the mid-point of

Also, is the mid-point of

And, is the mid-point of

Adding first three equations in i), ii) and iii) we get

Solving the first three equations, we get

Adding the next three equation in  i), ii) and iii) we get

Solving the next three equations in i) , ii) and iii) we get

Similarly, Adding the next three equation in  i), ii) and iii) we get

Solving the next three equations in i) , ii) and iii) we get

Therefore the vertices of the triangle are

 

Question 8: are the vertices of a triangle Find the point in which the bisector of the angle meets

Answer:

is the internal bisector of

Therefore divides in the ratio of

 

Question 9: Find the ratio in which the sphere divides the line joining the points

Answer:

Let the sphere meet the line joining the points at

Therefore we have

Let the point divide the line joining in the ratio

Substituting these values in equation i) we get

Therefore the sphere divides the line joining internally in the ratio of and externally in the ratio of

 

Question 10: Show that the plane divides the line joining the points in the ratio

Answer:

Let

Let the line joining and be divided by the plane at point in the ratio

Since lies on the given plane , it should satisfy it. Hence,

Therefore the plane divides the line joining the points

 

Question 11: Find the centroid of a triangle, mid-points of whose sides are

Answer:

Let   and   be the vertices of the given triangle.

And be the mid points of the sides respectively.

is the midpoint of

is the midpoint of

is the midpoint of

Adding equation i) , ii) and iii) we get

Therefore the coordinate of the centroid is given by

 

Question 12: The centroid of a triangle is at the point If the coordinates of are respectively, find the coordinates of the point .

Answer:

Let be the centroid of the

Given

Given

Let

 

Question 13: Find the coordinates of the points which trisect the line segment joining the points

Answer:

Given

Let and be the points that trisect

Therefore

Therefore

Thus divides internally in the ratio

Similarly,

Thus divides internally in the ratio