1. Find the value(s) of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.

2. Find the value of y for which the distance between the points A (3,-1) and B (11, y) is 10 units.

3. Point A(3, y) is equidistant from the points P(6, 5) and Q(0, -3). Find the value of y.

4. If point A(0, 2) is equidistant from points B(3, p) and C(p, 5), then find the value of p.

5. Find the value of k, if the point P(2, 4) is equidistant from the points A(5, k) and B(k, 7).

6. If the point P(k – 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k.

7. Find a point P on the y-axis which is equidistant from points A(4, 8) and B(-6, 6). Also, find the distance AP.

8. If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b), prove that by = ay.

9. If point A(0, 2) is equidistant from points B(3, p) and C(p, 5), find p. Also, find the length of the AB.

10. If the point P(2, 2) is equidistant from the points A(-2, k) and B(-2k, -3), find k. Also find the length of AP.

11.
Prove that the points (3, 0), (6, 4), and (-1, 3) are the vertices of a right-angled isosceles triangle.

12. Prove that the points A(0, -1), B(-2, 3), C(6, 7), and D(8, 3) are the vertices of a rectangle ABCD

13. Show that the points (-2, 3), (8, 3), and (6, 7) are the vertices of a right triangle.

14. Prove that the points A(2, 3), B(-2, 2), C(-1, -2), and D(3, -1) are the vertices of a square ABCD.

15. Prove that the points A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) are the vertices of a rhombus ABCD. Is ABCD a square?

16. Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) & D(2,6), are equal and bisect each other.

17. Find the value of k for which point (0, 2), is equidistant from two points (3, k) and (k, 5).

18. If the points A(-2, 1), B(a, b) and C(4, -1) are collinear and a – b = 1, find the values of a and b.

19. If the points A(-1, -4), B(b, c) and C(5, -1) are collinear and 2b + c = 4, find the values of b and c.

20. Find the area of the triangle ABC with A(1, 4) and mid-points of sides through A being (2, -1) and (0, -1).

21. Find the area of the triangle QPR with Q(3, 2) and the mid-points of the sides through Q being (2, -1) and (1, 2).

22. Find the ratio in which the y-axis divides the line segment joining the points (-4, -6) and (10, 12). Also, find the coordinates of the point of division

23. Find the ratio in which point P(-1, y) lies on the line segment joining points A(-3, 10) and B(6, -8) divides it. Also, find the value of y

24. In what ratio does point (14.4) divide the line segment joining the points P(2, -2) and Q(3, 7)? Also, find the value of y.

25. Find the coordinates of a point P on the line segment joining A(1, 2) and B(6, 7) such that AP = AB.

26. Points A(-1, y) and B(5, 7) lie on a circle with center 0(2, -3y). Find the values of y. Hence find the radius of the circle

27. If the points P(-3, 9), Q(a, b) and R(4, -5) are collinear and a + b = 1, find the values of a and b.

28. The area of a triangle is 5 sq. units. Two of its vertices are (2, 1) and (3,-2). If the third vertex is (7/2,y), find the value of y.

29. Find the ratio in which the line segment joining the points A(3, -3) and B(-2, 7) is divided by the x-axis. Also, find the coordinates of the point of division

30. In the Figure, ABC is a triangle coordinates of whose vertex A are (0, -1). D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the midpoint of BC, find the areas of ∆ABC and ∆DEF.

31. Find the distance of the point (-3, 4) from the x-axis

32.
If the points A(x, 2), B(-3, 4), and C(7, -5) are collinear, then find the value of

33. For what value of k will k + 9, 2k – 1, and 2k + 7 are the consecutive terms of an A.P.?

34. In which quadrant the point P that divides the line segment joining the points A(2, -5) and B(5,2) in the ratio 2 : 3 lies

35. ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3), and D(0, 3). Calculate the length of one of its diagonals.

36. In the figure, calculate the area of triangle ABC (in sq. units)