Distance of a point from a line and coordinates of triangle centres - One Line Questions

1. What are the coordinates of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃)? — ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
2. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the orthocenter. — (0, 0)
3. Find the orthocenter of a triangle with vertices (0, 0), (a, 0), and (0, b). — (0, 0)
4. Find the circumcenter of the triangle with vertices A(1, 0), B(-1, 0), and C(0, √3). — (0, 0)
5. The orthocenter of the triangle with vertices (0, 0), (1, 0), and (0, 1) is: — (0, 0)
6. Find the incenter of the triangle with vertices (0, 0), (3, 0), and (0, 4). — (1, 1)
7. Find the circumcenter of the triangle with vertices A(0, 0), B(2, 0), and C(0, 2). — (1, 1)
8. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the incenter. — (2, 2)
9. The coordinates of the vertices of a triangle are (3, 4), (5, -2), and (1, -6). Find the coordinates of the centroid. — (3, -4/3)
10. Find the centroid of the triangle with vertices A(1, 2), B(3, 4), and C(5, 6). — (3, 4)
11. For a triangle ABC, if A=(0,0), B=(3,0), C=(0,4), find the coordinates of its circumcenter. — (3/2, 2)
12. The coordinates of the vertices of a triangle are (1, 2), (3, 5), and (4, 1). Find the coordinates of its centroid. — (8/3, 8/3)
13. The circumcenter of a triangle with vertices (0, 0), (2a, 0), and (a, a√3) is: — (a, a/√3)
14. Find the orthocenter of a triangle with vertices A(0, 0), B(a, 0), and C(a/2, a√3/2). — (a/2, a√3/6)
15. Find the incenter of an equilateral triangle with vertices at (0, 0), (a, 0), and (a/2, a√3/2). — (a/2, a√3/6)
16. The circumcenter of a triangle with vertices (0, 0), (b, 0), and (0, c) is: — (b/2, c/2)
17. What is the formula for the distance of a point (x₁, y₁) from the line Ax + By + C = 0? — |Ax₁ + By₁ + C| / √(A² + B²)
18. The distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by: — |C₁ - C₂| / √(A² + B²)
19. If two lines are parallel, Ax + By + C₁ = 0 and Ax + By + C₂ = 0, what is the distance between them? — |C₁ - C₂| / √(A² + B²)
20. The distance of the point (a cosθ, a sinθ) from the line x cosθ + y sinθ - a = 0 is: — 0
21. Find the distance of the point (2, -1) from the line 2x + 5y + 6 = 0. — 0/√29
22. Find the distance of the point (3, 4) from the line 3x + 4y - 10 = 0. — 1
23. The distance of the point (2, 3) from the line x/3 + y/4 = 1 is: — 12/5
24. Find the distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0. — 15/5
25. What is the distance of the point (1, -2) from the line 3x - 4y + 5 = 0? — 18/5
26. What is the distance of the point P(2, 3) from the line y = 5? — 2
27. The centroid of a triangle divides each median in the ratio: — 2:1
28. Consider the line 5x - 12y + 26 = 0. Find the distance of the origin from this line. — 26/13
29. What is the distance of the point P(2, 3) from the line x = -1? — 3
30. Find the distance between the parallel lines y = 2x + 4 and y = 2x + 9. — 5/√5
31. Find the distance of the point (6, 8) from the line 4x + 3y - 12 = 0. — 60/5
32. If the distance of point P from the line √3x + 4y + 5 = 0 is 2, then P can be: — A point such that |3√3 + 8 + 5| = 10
33. If the centroid of a triangle with vertices (a, 1), (2, b), and (3, 4) is (2, 3), find a and b. — a=1, b=4
34. For a triangle ABC, the centroid G divides the median AD in the ratio: — AG:GD = 2:1
35. The incenter of a triangle is the intersection of: — Angle bisectors
36. The distance of the point (x, y) from the line Ax + By + C = 0 is proportional to: — Ax + By + C
37. In an equilateral triangle, which of the following points coincide? — Centroid, Orthocenter, Circumcenter, Incenter
38. The orthocenter, centroid, and circumcenter of a triangle are collinear. This line is known as the: — Euler line
39. What are the coordinates of the orthocenter of a triangle? — Intersection of altitudes
40. What are the coordinates of the incenter of a triangle? — Intersection of angle bisectors
41. What are the coordinates of the circumcenter of a triangle? — Intersection of perpendicular bisectors
42. If the distance of the point (1, k) from the line 3x + 4y - 12 = 0 is 3, find the value of k. — k = 0 or k = -12/2
43. If the distance of the point (k, 2) from the line 4x - 3y + 10 = 0 is 2 units, find the possible values of k. — k = 2 or k = -8
44. For a right-angled triangle, the circumcenter is the: — Midpoint of the hypotenuse
45. The line joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. This is a property related to: — Midpoint Theorem
46. The circumcenter of a triangle is the intersection of: — Perpendicular bisectors of the sides
47. The orthocenter of an isosceles triangle lies on: — The altitude to the base
48. The distance of the point (x₁, y₁) from the line Ax + By + C = 0 is positive if: — The point and the origin lie on opposite sides of the line
49. The distance of a point from a line is zero if and only if: — The point lies on the line
50. The incenter of a triangle is equidistant from: — The sides of the triangle
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