Question 1: Find the mid-point of the line segment joining the points :

(i) and

(ii) and

Answer:

i)

Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

ii)

Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

Question 2: Points have co-ordinates respectively. The mid-point of is . Find the values of .

Answer:

Given Midpoint of

Therefore

Therefore

Question 3: are the vertices of triangle . If is the mid-point of is the mid-point of , show that .

Answer:

Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

Similarly

Let the coordinates of the point

Therefore

Therefore

Length of

Length of

Hence

Question 4: Given is the mid-point of , find the co-ordinates of :

(i)

(ii)

Answer:

i)

Given Midpoint of and let

Therefore

Therefore

ii)

Given Midpoint of and let

Therefore

Therefore

Question 5: is the midpoint of line segment as shown in the given figure. Find the co-ordinates of Points .

Answer:

Given Midpoint of and let and

Therefore

Therefore

Question 6: In the given figure, is mid-point of line segment . Find the co-ordinates of .

Answer:

Given Midpoint of and let and

Therefore

Therefore

Question 7: are the vertices of a triangle. Find the length of its median though the vertex .

Answer:

Let . be the midpoint of .

Therefore

Therefore

Length of

Question 8: Given a line in which . Find the co-ordinates of .

Answer:

For

Ratio

Hence the coordinates of

For

Ratio

Hence the coordinates of

Question 9: One end of the diameter of a circle is Find the co-ordinates of the other end of it, if the center of the circle is .

Answer:

Let be the other end of the diameter

Hence the coordinates of the other point of the diameter is

Question 10: are the vertices of quadrilateral . Find the co-ordinates of the midpoint of . Give a special name to the quadrilateral.

Answer:

Let be the midpoint of .

Therefore

Let . be the midpoint of .

Therefore

Because the diagonals bisect each other, the quadrilateral is a Parallelogram.

Question 11: are the vertices of parallelogram are the co-ordinates of the point of intersection of its diagonals. Find co-ordinates of .

Answer:

Given Midpoint of and let

Therefore

Therefore

Given Midpoint of and let

Therefore

Therefore

Question 12: are the vertices of a parallelogram . Find the co-ordinates of vertex .

Answer:

Let be the midpoint of .

Therefore

Given Midpoint of and let

Therefore

Therefore

Question 13: The points are midpoints of the sides of a triangle. Find its vertices.

Answer:

Let the vertices of the triangle be

Given Midpoint of

Therefore

… … … … i)

… … … … ii)

Given Midpoint of

Therefore

… … … … iii)

… … … … iv)

Given Midpoint of

Therefore

… … … … v)

… … … … vi)

Adding i), iii) and v) we get

… … … … vii)

Adding ii), iv) and vi) we get

… … … … viii)

Using i), iii), v) and vii) we get

Similarly Using ii), iv), vi) and viii) we get

Hence the vertices are

Question 14: Points are collinear (i.e. lie on the same straight line) such that . Calculate the values of .

Answer:

Given is the midpoint of . Therefore

Therefore

Question 15: Points are collinear. If lies between , such that , calculate the values of .

Answer:

Given is the midpoint of . Therefore

Therefore

Question 16: Calculate the co-ordinates of the centroid of the triangle , if .

Answer:

Let be the centroid of triangle .

Therefore

Hence the coordinates of the centroid are

Question 17: The co-ordinates of the centroid of a triangle are . lf ; calculate the co-ordinates of vertex .

Answer:

Given be the centroid of triangle .

Therefore

Hence the coordinates of P are

Question 18: are the vertices of the triangle whose centroid is the origin. Calculate the values of .

Answer:

Given be the centroid of triangle .

Therefore

Hence