Question 1: Calculate the distance between the points and correct to decimal places.

Answer:

and

Distance

Question 2: Find the distance between the points and correct to significant figures.

Answer:

and

Distance

Question 3: Show that the points form an equilateral triangle.

Answer:

Therefore three sides are equal which makes it an equilateral triangle.

Question 4: The circle with center passes though the points . Find the values of .

Answer:

Distance of the points from the center are equal. Therefore

… … … … i)

… … … … ii)

Solving i) and ii), we get and .

Therefore the center is

Question 5: The points are such that . Find a linear relation between .

Answer:

Question 6: Given a triangle in which . A point lies on such that . Find the length of line segment .

Answer:

A point lies on such that .

Therefore

Therefore

Length of

Question 7: are two fixed points. Find the co-ordinates of the point in such that: . Also, find the co-ordinates of some other point such that .

Answer:

For P When Ratio:

Therefore

Therefore the point

For Q When Ratio:

Therefore

Therefore the point

Question 8: are the vertices of a triangle . Point lies on lies on such that . Show that: .

Answer:

For When Ratio:

Therefore

Therefore the point

For When Ratio:

Therefore

Therefore the point

Therefore

Which proves that

Question 9: Find the co-ordinates of points of trisection of the line segment joining the point and the origin.

Answer:

Let be the two points dividing the points and the origin in the ratio 1:2 and 2:1 respectively.

Therefore for

Hence

Therefore for

Hence

Question 10: 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 11: 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 12: The mid-point of the segment , as shown in diagram, is . Write down the coordinates of . [1996]

Answer:

Given Midpoint of

Therefore

Therefore

Question 13: 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 14: 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 15: The mid-point of the line segment joining . Find the values of .

Answer:

Given Midpoint of

Therefore

Question 16: The mid-point of the line segment joining . Find the values of . [2007]

Answer:

Given Midpoint of

Therefore

Question 17: (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 18: 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 19: is the mid-point of the line segment joining the points . Find the coordinates of point . Further, if divides the line segment joining and the origin in the ratio , find the ratio .

Answer:

For M When Ratio: for

Therefore

Therefore the point

Let divide MO in the ratio

Since

Question 20: 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