Question 1: Find the the distance between the points (3, 6) and (0,2).

Answer:

Therefore the distance between the given points

Question 2: Find the distance between the origin and the point : (i) (-12, 5) (ii) (15, -8).

Answer:

Question 3: Find the co-ordinates of points on the x-axis which are at a distance of 5 units from the point (6, -3).

Answer:

Let the co-ordinates of the point on the x-axis be

Squaring both sides

Therefore the required points on the x-axis are and

Question 4: KM is a straight line of 13 units. If K has the coordinate ( 2, 5) and M has the coordinates , find the value of . [ICSE Board 2004]

Answer:

unit

Squaring both sides

Question 5: Which point on the y-axis is equidistant from the points (12,3) and (-5, 10).

Answer:

Let the required point on the y-axis be .

Given is equidistant from and

i.e. distance between and distance between and

Therefore the required point on the axis is

Question 6: Use the distance formula to show that the points A(1, 1), B(6, 4) and C(4,2) are collinear.

Answer:

Therefore given points are collinear.

Question 7: Show that the points A (8, 3), B (0, 9) and C (14, 11) are the vertices of an isosceles right-angled triangle.

Answer:

the triangle is right-angled triangle.

and, the triangle is isosceles.

Hence, the is an isosceles right-angled triangle.

Question 8: Show that the quadrilateral ABCD with A (3, 1), B (0, -2), C (1, 1) and D (4, 4) is a parallelogram.

Answer:

Since the opposite sides of the quadrilateral ABCD are equal, it is a parallelogram.

Question 9: Find the area of a circle, whose center is (5, -3) and which passes through the point ( -7, 2). Take

Answer:

The radius of the circle = distance between the points and

Therefore Area of the circle sq. units

Question 10: Find the points on the x-axis whose distances from the points A(7, 6) and B(-3, 4) are in the ratio 1 : 2.

Answer:

Let the required point on x-axis

Question 11: Point is equidistant from the points A(-2, 0) and B(3, – 4). prove that : .

Answer:

Question 12: Find the co-ordinates of the circumcenter of the triangle ABC; whose vertices A, B and C are (4,6), (0,4) and (6, 2) respectively.

Answer:

Let the circumcenter be

Therefore

and

On solving i) and ii), we get : and

Therefore the circumcenter of the given triangle

Question 13: Find the co-ordinates of point P which divides the join of A (4, -5) and B(6, 3) in the ratio 2 : 5.

Answer:

Let the co-ordinates of be

Question 14: Find the ratio in which the point (5, 4) divides the line joining points (2, 1) and (7, 6)

Answer:

Let the required ratio be

Take and

Therefore the required ratio is

Question 15: In what ratio is the line joining the points (4, 2) and (3, -5) divided by the x-axis ? Also, find the co-ordinates of the point of intersection.

Answer:

Let the required ratio be and the point on the x-axis be

Question 16: Calculate the ratio in which the line joining the points (4, 6) and (-5, -4) is divided by the line y = 3. Also, find the co-ordinates of the point of intersection.

Answer:

Question 17: The origin O, B (-6, 9) and C (12, -3) are vertices of triangle OBC. Point P divides OB in the ratio 1 : 2 and point Q divides OC in the ratio 1 : 2. Find the co-ordinates of points P and Q. Also, show that

Answer:

Question 18: Find the coordinates of the points of trisection of the line segment joining the points A (6, -2) and B (-8, 10).

Answer:

For P:

For Q:

Question 19: Show that is a point of trisection of the line segment joining points A (4, -2) and B (1, 4). Hence, find the value of .

Answer:

will be a point of trisection of if it divides in the ratio or

Hence is a point of trisection of

Question 20: Find the co-ordinates of the mid-point of the line segment joining the points P(4, -6) and Q(-2, 4).

Answer:

Question 21: The mid-point of line segment AB (shown in the diagram) is (-3, 5). Find the co-ordinates of A and B.

Answer:

Since, point lies on the x-axis; let

Since, point lies on the y-axis; let

Co-ordinates of and co-ordinates of

Question 22: A(14, -2), B(6, -2) and D (8, 2) are the three vertices of a parallelogram ABCD. Find the coordinates of the fourth vertex C.

Answer:

Let

Since the diagonals of a parallelogram bisect each other;

Mid-point of AC = mid-point of BD

Therefore the vertex is

Question 23: In triangle ABC, P (-2,5) is mid-point of AB, Q (2, 4) is mid-point of BC and R (-1, 2) is mid-point of AC. Calculate the co-ordinates of vertices A, B and C.

Answer:

Adding equations i) , iii) and v) we get

Subtracting equation i) from vii) we get

Subtracting equation iii) from vii) we get

Subtracting equation v) from vii) we get

Adding equations ii) , iv) and vi) we get

Subtracting equation ii) from viii) we get

Subtracting equation iv) from viii) we get

Subtracting equation vi) from viii) we get

Question 24: The mid-point of the line segment joining (3m, 6) and (- 4,3n) is (1,2m -1). Find the values of m and n. [ICSE Board 2006]

Answer:

According to the adjoining figure, we have :

Question 25: Find the co-ordinates of the point of intersection of the medians of triangle ABC; given A = (-2,3), B = (6, 7) and C = (4, 1).

Answer:

Let be the mid-point of

If is the point of intersection of medians (centroid), it divides the median in the ratio

Alternate way:

Question 26: ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B=(4, b) and C = (a,1), find ‘a’ and ‘b’. Find the length of side BC. [ICSE Board 2011]

Answer:

Since, is the centroid of

Therefore

and

units