*Note: We will use the formula. If a point divides two points and in the ratio , then the coordinates of the point at *

Question 1: Calculate the co-ordinates of the point which divides the line segment joining:

(i) and in the ratio

(ii) and in the ratio

Answer:

i)

Ratio:

Let the coordinates of the point

Therefore

Therefore

ii)

Ratio:

Let the coordinates of the point

Therefore

Therefore

Question 2: In what ratio is the line joining and divided by the .

Answer:

Let the required ratio be and the point of be

Since

Question 3: In what ratio is the line joining and divided by the .

Answer:

Let the required ratio be and the point of be

Since

Question 4: In what ratio does the point divide the join of and ? Also, find the value of .

Answer:

Let the point divide the join of and in the ratio

Since

Now calculate

Question 5: In what ratio does the point divide the join of and ? Also, find the value of .

Answer:

Let the point divide the join of and in the ratio

Since

Now calculate

Question 6: In what ratio is the join of and divided by the . Also, find the co-ordinates of the point of intersection.

Answer:

Let the required ratio be and the point of be

Since

Now calculate the coordinate of the point of intersection

Co-ordinates of the point of intersection =

Question 7: Find the ratio in which the join of and is divided by the . Also, find the co-ordinates of the point of intersection.

Answer:

Let the required ratio be and the point of be

Since

Now calculate the coordinate of the point of intersection

Co-ordinates of the point of intersection =

Question 8: Points divide the line segment joining the point and the origin in five equal parts. Find the co-ordinates of .

Answer:

For

Ratio

Hence the coordinates of

For

Ratio

Hence the coordinates of

Question 9: The line joining the points and is divided by the point such that Find the co-ordinates of .

Answer:

Given

This implies that

Hence the coordinates of the point

Question 10: is a point on the line joining and such that Find the coordinates of .

Answer:

Given

This implies

This implies that

Hence the coordinates of the point

Question 11: Calculate the ratio in which the line joining the points and is divided by the line . Also, find the co-ordinates of the point of intersection.

Answer:

Let the required ratio be and the point of be

Since

Now calculate the coordinate of the point of intersection

Co-ordinates of the point of intersection =

Question 12: Calculate the ratio in which the line joining and is divided by the line . [2006]

Answer:

Let the required ratio be and the point of be

Since

Now calculate the coordinate of the point of intersection

Co-ordinates of the point of intersection =

Question 13: The point divides the line segment , as shown in the figure, in the ratio . Find the co-ordinates of points .

Answer:

Ratio:

Therefore

Therefore

Question 14: Find the co-ordinates of the points of trisection of the line joining the points and .

Answer:

When Ratio:

Therefore

Therefore the point

When Ratio:

Therefore

Therefore the point

Question 15: Show that the Line segment joining the points and is trisection by the co-ordinate axes.

Answer:

Let the two points trisecting the points and are .

When Ratio for B:

Therefore

Therefore the point

When Ratio for A:

Therefore

Therefore the point

Question 16: Show that is a point of trisection of the line-segment joining the points and . Also find the co-ordinates of the other point of trisection.

Answer:

Let the point divide the join of and in the ratio

Since

For the other point

When Ratio:

Therefore

Therefore the point

Question 17: lf and

(i) find the length of

(ii) In what ratio is the line joining , divided by the ? [2008]

Answer:

Let the required ratio be and the point of be

Since

Question 18: The line segment joining the points and is intersected by the at point . Write down the abscissa of . Hence, find the ratio in which divides . Also, find the co-ordinates of .

Answer:

Let the required ratio be and the point of be

Since

Therefore the point

Question 19: are the co-ordinates of the vertices of the triangle . Points lie on and respectively, such that:

(i) Calculate the co-ordinates of and .

(ii) Show that

Answer:

For P When Ratio:

Therefore

Therefore the point

For Q When Ratio:

Therefore

Therefore the point

Question 20: are the vertices of a triangle . Find the length of line segment , where point lies inside , such that

Answer:

For P When Ratio:

Therefore

Therefore the point

Question 21: The line segment joining and is intercepted by at the point . Write down the ordinate of the point . Hence, find the ratio in which divides . Also, find the co-ordinates of the point . [1990, 2006]

Answer:

Let the required ratio be and the point of be

Since

Therefore the point

Question 22: The line segment joining and is intercepted by the at the point . Write down the abscissa of the point . Hence, find the ratio in which divides . Also, find the co-ordinates of the point .

Answer:

Let the required ratio be and the point of be

Since

Therefore the point

Question 23: The line joining and intersects the at point . and are perpendiculars from on the . Find:

(i) The ratio .

(ii) The co-ordinates of .

(iii) The area of the quadrilateral .

Answer:

Let the required ratio be and the point of be

Since

Therefore the point

Area of quadrilateral sq. units

Question 24: In the given figure, line meets the at point and at point . is the point and Find the co-ordinates of . [1999, 2013]

Answer:

Given

Therefore

Therefore .

Question 25: Given a line segment joining the points and . Find:

- i) The ratio in which is divided by .
- ii) Find the coordinates of point of intersection

iii) The length of [2012]

Answer:

Let the required ratio be and the point of intersection be

Since

Therefore the point intersection is

Length of .