Distance of a point from a line and coordinates of triangle centres - Question Bank

1. The orthocenter of an isosceles triangle lies on:
A) The altitude to the base
B) The median to the base
C) The perpendicular bisector of the base
D) All altitudes
2. For a triangle ABC, the centroid G divides the median AD in the ratio:
A) AG:GD = 2:1
B) AG:GD = 1:2
C) AG:GD = 1:1
D) AG:GD = 3:1
3. The distance of the point (x, y) from the line Ax + By + C = 0 is proportional to:
A) Ax + By + C
B) (Ax + By + C)²
C) √(Ax + By + C)
D) 1 / (Ax + By + C)
4. If the distance of the point (1, k) from the line 3x + 4y - 12 = 0 is 3, find the value of k.
A) k = 0 or k = -12/2
B) k = 3 or k = -9
C) k = 0 or k = -6
D) k = 1 or k = -5
5. The distance of the point (2, 3) from the line x/3 + y/4 = 1 is:
A) 12/5
B) 5/12
C) 1
D) 0
6. Find the incenter of the triangle with vertices (0, 0), (3, 0), and (0, 4).
A) (1, 1)
B) (2, 2)
C) (3, 4)
D) (0, 0)
7. The circumcenter of a triangle with vertices (0, 0), (b, 0), and (0, c) is:
A) (b/2, c/2)
B) (0, 0)
C) (b, c)
D) (b/2, 0)
8. Find the orthocenter of a triangle with vertices A(0, 0), B(a, 0), and C(a/2, a√3/2).
A) (a/2, a√3/6)
B) (0, 0)
C) (a, 0)
D) (a/2, 0)
9. The coordinates of the vertices of a triangle are (3, 4), (5, -2), and (1, -6). Find the coordinates of the centroid.
A) (3, -4/3)
B) (9, -4)
C) (3, 0)
D) (4, 1)
10. 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:
A) Midpoint Theorem
B) Centroid Theorem
C) Pythagorean Theorem
D) Sine Rule
11. If the distance of point P from the line √3x + 4y + 5 = 0 is 2, then P can be:
A) A point such that |3√3 + 8 + 5| = 10
B) A point such that |3√3 + 8 + 5| = 20
C) A point such that |3√3 + 8 + 5| = 5
D) A point such that |3√3 + 8 + 5| = 2
12. The distance of the point (a cosθ, a sinθ) from the line x cosθ + y sinθ - a = 0 is:
A) 0
B) a
C) 2a
D) a/2
13. Find the distance of the point (2, -1) from the line 2x + 5y + 6 = 0.
A) 0/√29
B) √29/0
C) 10/√29
D) 2/√29
14. The circumcenter of a triangle with vertices (0, 0), (2a, 0), and (a, a√3) is:
A) (a, a/√3)
B) (a, 0)
C) (0, 0)
D) (a, a)
15. If the orthocenter of a triangle with vertices (0, 0), (a, 0), and (0, b) is (0, 0), then:
A) The triangle is right-angled at the origin
B) The triangle is equilateral
C) The triangle is isosceles
D) The triangle is scalene
16. The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is G. If G = (0,0), then:
A) x₁ + x₂ + x₃ = 0 and y₁ + y₂ + y₃ = 0
B) x₁ + x₂ + x₃ = 3 and y₁ + y₂ + y₃ = 3
C) x₁ + x₂ + x₃ = 1 and y₁ + y₂ + y₃ = 1
D) x₁x₂x₃ = 0 and y₁y₂y₃ = 0
17. Find the distance between the parallel lines y = 2x + 4 and y = 2x + 9.
A) 5/√5
B) √5/5
C) 5/2
D) 2/5
18. The distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by:
A) |C₁ - C₂| / √(A² + B²)
B) |C₁ + C₂| / √(A² + B²)
C) |C₁ - C₂| / (A² + B²)
D) √(C₁² - C₂²) / √(A² + B²)
19. Find the incenter of an equilateral triangle with vertices at (0, 0), (a, 0), and (a/2, a√3/2).
A) (a/2, a√3/6)
B) (a/2, a√3/2)
C) (0, 0)
D) (a, 0)
20. For a triangle ABC, if A=(0,0), B=(3,0), C=(0,4), find the coordinates of its circumcenter.
A) (3/2, 2)
B) (1, 1)
C) (0, 0)
D) (2, 3)
21. The orthocenter of the triangle with vertices (0, 0), (1, 0), and (0, 1) is:
A) (0, 0)
B) (1, 1)
C) (1/2, 1/2)
D) (0, 1)
22. What is the distance of the point (1, -2) from the line 3x - 4y + 5 = 0?
A) 18/5
B) 5/18
C) 18/13
D) 13/18
23. Find the distance of the point (6, 8) from the line 4x + 3y - 12 = 0.
A) 60/5
B) 5/60
C) 12/5
D) 60/13
24. The distance of the point (x₁, y₁) from the line Ax + By + C = 0 is positive if:
A) The point and the origin lie on opposite sides of the line
B) The point and the origin lie on the same side of the line
C) The line passes through the origin
D) The line is horizontal
25. The incenter of a triangle is the intersection of:
A) Angle bisectors
B) Perpendicular bisectors
C) Medians
D) Altitudes
26. Find the circumcenter of the triangle with vertices A(1, 0), B(-1, 0), and C(0, √3).
A) (0, 0)
B) (1, 1)
C) (0, 1)
D) (1, 0)
27. The circumcenter of a triangle is the intersection of:
A) Perpendicular bisectors of the sides
B) Medians
C) Altitudes
D) Angle bisectors
28. If the centroid of a triangle with vertices (a, 1), (2, b), and (3, 4) is (2, 3), find a and b.
A) a=1, b=4
B) a=2, b=3
C) a=3, b=2
D) a=4, b=1
29. The coordinates of the vertices of a triangle are (1, 2), (3, 5), and (4, 1). Find the coordinates of its centroid.
A) (8/3, 8/3)
B) (2, 3)
C) (3, 4)
D) (4, 2)
30. What is the distance of the point P(2, 3) from the line x = -1?
A) 3
B) 2
C) 1
D) 0
31. What is the distance of the point P(2, 3) from the line y = 5?
A) 2
B) 3
C) 5
D) 0
32. Find the orthocenter of a triangle with vertices (0, 0), (a, 0), and (0, b).
A) (0, 0)
B) (a, b)
C) (a/2, b/2)
D) (a, 0)
33. For a right-angled triangle, the circumcenter is the:
A) Midpoint of the hypotenuse
B) Vertex with the right angle
C) Intersection of the altitudes
D) Centroid
34. In an equilateral triangle, which of the following points coincide?
A) Centroid, Orthocenter, Circumcenter, Incenter
B) Centroid and Orthocenter only
C) Circumcenter and Incenter only
D) Orthocenter and Centroid only
35. The orthocenter, centroid, and circumcenter of a triangle are collinear. This line is known as the:
A) Euler line
B) Median line
C) Altitude line
D) Bisector line
36. Find the distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0.
A) 15/5
B) 5/15
C) 15/25
D) 25/15
37. If two lines are parallel, Ax + By + C₁ = 0 and Ax + By + C₂ = 0, what is the distance between them?
A) |C₁ - C₂| / √(A² + B²)
B) |C₁ + C₂| / √(A² + B²)
C) |C₁ - C₂| / (A² + B²)
D) |C₁ - C₂| / |A + B|
38. Consider the line 5x - 12y + 26 = 0. Find the distance of the origin from this line.
A) 26/13
B) 13/26
C) 26/5
D) 12/26
39. The distance of a point from a line is zero if and only if:
A) The point lies on the line
B) The point is the origin
C) The line passes through the origin
D) The line is parallel to the x-axis
40. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the incenter.
A) (2, 2)
B) (1, 1)
C) (3, 4)
D) (0, 0)
41. The incenter of a triangle is equidistant from:
A) The sides of the triangle
B) The vertices of the triangle
C) The medians of the triangle
D) The altitudes of the triangle
42. What are the coordinates of the incenter of a triangle?
A) Intersection of angle bisectors
B) Intersection of altitudes
C) Intersection of medians
D) Intersection of perpendicular bisectors
43. Find the circumcenter of the triangle with vertices A(0, 0), B(2, 0), and C(0, 2).
A) (1, 1)
B) (0, 0)
C) (2, 2)
D) (1, 0)
44. What are the coordinates of the circumcenter of a triangle?
A) Intersection of perpendicular bisectors
B) Intersection of altitudes
C) Intersection of medians
D) Intersection of angle bisectors
45. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the orthocenter.
A) (0, 0)
B) (1, 1)
C) (2, 2)
D) (3, 4)
46. What are the coordinates of the orthocenter of a triangle?
A) Intersection of altitudes
B) Intersection of medians
C) Intersection of perpendicular bisectors
D) Intersection of angle bisectors
47. The centroid of a triangle divides each median in the ratio:
A) 2:1
B) 1:2
C) 1:1
D) 3:1
48. Find the centroid of the triangle with vertices A(1, 2), B(3, 4), and C(5, 6).
A) (3, 4)
B) (4, 5)
C) (5, 4)
D) (3, 3)
49. What are the coordinates of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃)?
A) ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
B) ((x₁ + x₂ + x₃)/2, (y₁ + y₂ + y₃)/2)
C) ((x₁ - x₂ - x₃)/3, (y₁ - y₂ - y₃)/3)
D) ((x₁x₂x₃)/3, (y₁y₂y₃)/3)
50. If the distance of the point (k, 2) from the line 4x - 3y + 10 = 0 is 2 units, find the possible values of k.
A) k = 2 or k = -8
B) k = -2 or k = 8
C) k = 2 or k = 8
D) k = -2 or k = -8
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