Question 1: A line segment joining is divided in the ratio , the point where the line segment intersects the .

(i) Calculate the value of

(ii) Calculate the co-ordinates of . [1994]

Answer:

Therefore for

Hence

Question 2: In what ratio is the line joining divided by the ? Write the co-ordinates of the point where intersects the . [1993]

Answer:

Let the required ratio be and the point of be

Since

Therefore

Therefore

Question 3: The mid-point of the segment , as shown in diagram, is . Write down the coordinates of . [1996]

Answer:

Given Midpoint of

Therefore

Therefore

Question 4: is a diameter of a circle with center . If , find

(i) the length of radius

(ii) the coordinates of . [2013]

Answer:

Given Midpoint of

Therefore

Therefore

Question 5: Find the co-ordinates of the centroid of a triangle whose vertices are : . [2006]

Answer:

Let be the centroid of triangle .

Therefore

Hence the coordinates of the centroid are

Question 6: The mid-point of the line segment joining . Find the values of . [2007]

Answer:

Given Midpoint of

Therefore

Question 7: (i) Write down the co-ordinates of the point that divides the line joining in the ratio .

(ii) Calculate the distance , where is the origin.

(iii) In what ratio does the divide the line ? [1995]

Answer:

i) For P When Ratio:

Therefore

Therefore the point

ii)

iii) Let the required ratio be and the point be

Since

Question 8: Prove that the points are the vertices of an isosceles right-angled triangle. Find the co-ordinates of so that is a square. [1992]

Answer:

Since (two sides are equal). Hence triangle is a isosceles triangle.

Question 9: Calculate the ratio in which the line joining is divided by point . Also, find (i) (ii) length of . [2014]

Answer:

Let divide MO in the ratio

Since

Since

Question 10: 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 11: 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 12: 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 13: 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 14: 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 .

Question 15: is a straight line of units. If has the coordinates and has the coordinates , find the value of . [2004]

Answer:

and are the two points.

Distance between them is units.

Therefore

Question 16: The mid point of the line segment joining (3m, 6) and (-4, 3n) is (1, 2m-1). Find the values of m and n. [2006]

Answer:

Given Midpoint of

Therefore

Question 17: is a triangle and is the centroid of the triangle. If , find . Find the length of the side . [2011]

Answer:

Since is the centroid

Therefore

Therefore units.