Section formula in three dimensions - Question Bank

1. If the point (3, 4, 5) divides the line segment joining A(a, b, c) and B(d, e, f) in the ratio 1:2 internally, then which equation is correct?
A) 3 = (2a + d)/3
B) 4 = (2b + e)/3
C) 5 = (2c + f)/3
D) All of the above
2. Find the coordinates of the point that divides the line joining A(1, 2, 3) and B(4, 5, 6) externally in the ratio 2:1.
A) (7, 8, 9)
B) (9, 8, 7)
C) (8, 7, 9)
D) (8, 9, 7)
3. In what ratio does the point (1, 2, 3) divide the line segment joining A(2, 1, -1) and B(3, 0, 2)?
A) 2:1
B) 1:2
C) 3:1
D) 1:3
4. Find the coordinates of the point that divides the line segment joining A(1, -2, 3) and B(3, 4, -5) internally in the ratio 1:2.
A) (5/3, 0, 1/3)
B) (1/3, 0, 5/3)
C) (0, 5/3, 1/3)
D) (0, 1/3, 5/3)
5. If point P divides the line segment joining A(2, 3, 4) and B(5, 6, 7) in the ratio 1:2 externally, find the coordinates of P.
A) (-1, 0, -1)
B) (0, -1, -1)
C) (-1, -1, 0)
D) (1, 1, 1)
6. Find the coordinates of the point that divides the line segment joining A(1, 2, 3) and B(3, 4, 5) in the ratio 3:2 internally.
A) (17/5, 21/5, 25/5)
B) (25/5, 21/5, 17/5)
C) (21/5, 17/5, 25/5)
D) (17/5, 25/5, 21/5)
7. If the point (1, 2, 3) divides the line segment joining A(a, b, c) and B(d, e, f) externally in the ratio 1:2, then which of the following is true?
A) 2a - d = 1, 2b - e = 2, 2c - f = 3
B) a - 2d = 1, b - 2e = 2, c - 2f = 3
C) d - 2a = 1, e - 2b = 2, f - 2c = 3
D) 2d - a = 1, 2e - b = 2, 2f - c = 3
8. Find the coordinates of the point that divides the line joining A(1, 1, 1) and B(3, 3, 3) externally in the ratio 1:2.
A) (-1, -1, -1)
B) (1, 1, 1)
C) (3, 3, 3)
D) (0, 0, 0)
9. In what ratio does the point (2, 3, 4) divide the line segment joining A(1, 0, -1) and B(3, 6, 9)?
A) 1:1
B) 2:1
C) 1:2
D) 3:1
10. Find the coordinates of the point that divides the line segment joining A(1, 2, 3) and B(2, 3, 4) externally in the ratio 3:2.
A) (4, 1, -1)
B) (1, -1, 4)
C) (-1, 4, 1)
D) (1, 4, -1)
11. If point P divides the line segment AB joining A(3, -1, 2) and B(5, 3, 4) in the ratio 1:2 internally, find the coordinates of P.
A) (17/3, 1/3, 8/3)
B) (8/3, 17/3, 1/3)
C) (1/3, 8/3, 17/3)
D) (17/3, 8/3, 1/3)
12. Find the coordinates of the point that divides the line joining A(2, 1, -1) and B(1, 2, 3) internally in the ratio 2:1.
A) (4/3, 5/3, 1/3)
B) (5/3, 4/3, 1/3)
C) (1/3, 5/3, 4/3)
D) (4/3, 1/3, 5/3)
13. If the point (1, 2, 3) divides the line segment joining A(a, b, c) and B(d, e, f) internally in the ratio 2:1, then which of the following is true?
A) 2d + a = 3, 2e + b = 6, 2f + c = 9
B) d + 2a = 3, e + 2b = 6, f + 2c = 9
C) 2d + 2a = 3, 2e + 2b = 6, 2f + 2c = 9
D) d + a = 3, e + b = 6, f + c = 9
14. Find the coordinates of the point that divides the line segment joining A(1, 2, 3) and B(3, 4, 5) externally in the ratio 1:3.
A) (-3/2, -1, -1/2)
B) (-1/2, -3/2, -1)
C) (-1, -3/2, -1/2)
D) (-3/2, -1/2, -1)
15. In what ratio does the point (1, 1, 1) divide the line segment joining A(2, 3, 4) and B(0, -1, -2)?
A) 1:1
B) 2:1
C) 1:2
D) 3:1
16. Find the coordinates of the point that divides the line segment joining A(2, -1, 3) and B(1, 2, -1) externally in the ratio 1:2.
A) (0, -7, 7)
B) (7, 0, -7)
C) (0, 7, -7)
D) (7, -7, 0)
17. If the point P divides the line segment joining A(x1, y1, z1) and B(x2, y2, z2) internally in the ratio m:n, and P is the origin (0,0,0), then which of the following is true?
A) mx2 + nx1 = 0, my2 + ny1 = 0, mz2 + nz1 = 0
B) mx2 - nx1 = 0, my2 - ny1 = 0, mz2 - nz1 = 0
C) nx2 + mx1 = 0, ny2 + my1 = 0, nz2 + mz1 = 0
D) nx2 - mx1 = 0, ny2 - my1 = 0, nz2 - mz1 = 0
18. Find the coordinates of the point that divides the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 3:1 internally.
A) (13/4, 17/4, 7)
B) (7, 17/4, 13/4)
C) (13/4, 7, 17/4)
D) (17/4, 13/4, 7)
19. If the point (3, 4, 5) divides the line segment joining A(a, b, c) and B(d, e, f) in the ratio 2:1 externally, then which of the following is true?
A) a - 2d = 9, b - 2e = 12, c - 2f = 15
B) 2d - a = 9, 2e - b = 12, 2f - c = 15
C) d - 2a = 9, e - 2b = 12, f - 2c = 15
D) 2a - d = 9, 2b - e = 12, 2c - f = 15
20. Find the coordinates of the point that divides the line segment joining A(1, 1, 1) and B(2, 2, 2) externally in the ratio 2:1.
A) (0, 0, 0)
B) (1, 1, 1)
C) (2, 2, 2)
D) (3, 3, 3)
21. In what ratio does the point (2, 4, 6) divide the line segment joining A(1, 2, 3) and B(3, 6, 9)?
A) 1:1
B) 2:1
C) 1:2
D) 3:1
22. A point divides the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 1:2 externally. Find the coordinates of this point.
A) (2, 1, 0)
B) (0, 1, 2)
C) (1, 2, 0)
D) (0, 2, 1)
23. Find the coordinates of the point which divides the line segment joining A(3, 4, 5) and B(1, 2, 3) in the ratio 1:2 externally.
A) (5, 6, 7)
B) (7, 6, 5)
C) (5, 7, 6)
D) (6, 5, 7)
24. If a point P divides the line segment joining A(2, 3, 4) and B(5, 6, 7) externally in the ratio 1:2, find the coordinates of P.
A) (-1, 0, -1)
B) (0, -1, -1)
C) (-1, -1, 0)
D) (0, 0, 0)
25. Find the coordinates of the point that divides the line segment joining A(1, 2, 3) and B(3, 4, 5) internally in the ratio 2:1.
A) (7/3, 8/3, 10/3)
B) (10/3, 8/3, 7/3)
C) (8/3, 7/3, 10/3)
D) (8/3, 10/3, 7/3)
26. If the point (x, y, z) is the centroid of the tetrahedron formed by the vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3), and D(x4, y4, z4), what are the coordinates of the centroid?
A) ((x1+x2+x3+x4)/4, (y1+y2+y3+y4)/4, (z1+z2+z3+z4)/4)
B) ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)
C) ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
D) ((x1+x2+x3+x4)/3, (y1+y2+y3+y4)/3, (z1+z2+z3+z4)/3)
27. Find the coordinates of the point that divides the line joining A(0, 0, 0) and B(1, 2, 3) internally in the ratio 1:1.
A) (1/2, 1, 3/2)
B) (1, 2, 3)
C) (0, 0, 0)
D) (1, 1, 1)
28. If the point P(1, 2, 3) divides the line segment joining A(a, b, c) and B(d, e, f) externally in the ratio 2:1, then which of the following is true?
A) a - 2d = 2, b - 2e = 4, c - 2f = 6
B) 2d - a = 2, 2e - b = 4, 2f - c = 6
C) d - 2a = 2, e - 2b = 4, f - 2c = 6
D) 2a - d = 2, 2b - e = 4, 2c - f = 6
29. Find the coordinates of the point which divides the line segment joining A(2, 3, 4) and B(1, 2, 3) in the ratio 2:3 internally.
A) (8/5, 11/5, 13/5)
B) (13/5, 11/5, 8/5)
C) (11/5, 13/5, 8/5)
D) (8/5, 13/5, 11/5)
30. In what ratio does the point (1, 2, 3) divide the line segment joining A(2, 1, -1) and B(3, -1, 2)?
A) 2:1
B) 1:2
C) 3:1
D) 1:3
31. Find the coordinates of the point that divides the line joining A(5, -2, 3) and B(7, 4, -1) externally in the ratio 2:3.
A) (-1, -22, 11)
B) (11, -22, -1)
C) (-22, 11, -1)
D) (-1, 11, -22)
32. A point P divides the line segment joining A(1, -2, 3) and B(3, 4, -5) in the ratio 2:3 internally. Find the coordinates of P.
A) (17/5, 4/5, 3/5)
B) (3/5, 4/5, 17/5)
C) (4/5, 17/5, 3/5)
D) (4/5, 3/5, 17/5)
33. If the point (3, 4, 5) divides the line segment joining A(a, b, c) and B(d, e, f) in the ratio 1:2 internally, then which of the following is true?
A) d + 2a = 9, e + 2b = 12, f + 2c = 15
B) 2d + a = 9, 2e + b = 12, 2f + c = 15
C) d + a = 9, e + b = 12, f + c = 15
D) 2d + 2a = 9, 2e + 2b = 12, 2f + 2c = 15
34. Find the coordinates of the point that divides the line segment joining A(2, 4, 6) and B(1, 3, 2) externally in the ratio 3:2.
A) (4, 6, 14)
B) (14, 4, 6)
C) (6, 14, 4)
D) (4, 14, 6)
35. A point divides the line segment joining A(1, 1, 1) and B(2, 2, 2) in the ratio 1:2 internally. What are the coordinates of this point?
A) (1, 1, 1)
B) (4/3, 4/3, 4/3)
C) (5/3, 5/3, 5/3)
D) (2, 2, 2)
36. If P(2, -1, 3) divides the line segment joining A(x1, y1, z1) and B(4, 5, 6) internally in the ratio 1:2, find the coordinates of A.
A) (0, -5, 7)
B) (7, -5, 0)
C) (0, 7, -5)
D) (-5, 0, 7)
37. Find the coordinates of the point that divides the line segment joining P(3, -2, 1) and Q(6, 1, -2) in the ratio 2:1 internally.
A) (5, 0, -1/3)
B) (5, 0, -1)
C) (5, -1, 0)
D) (0, 5, -1)
38. The point P divides the line joining A(x1, y1, z1) and B(x2, y2, z2) externally in the ratio m:n. If P is the origin (0,0,0), then which of the following is true?
A) mx2 + nx1 = 0, my2 + ny1 = 0, mz2 + nz1 = 0
B) mx2 - nx1 = 0, my2 - ny1 = 0, mz2 - nz1 = 0
C) nx2 + mx1 = 0, ny2 + my1 = 0, nz2 + mz1 = 0
D) nx2 - mx1 = 0, ny2 - my1 = 0, nz2 - mz1 = 0
39. If the point P(a, 2, -3) divides the line segment joining A(3, -1, 4) and B(5, 3, 6) internally in the ratio 1:2, find the value of 'a'.
A) 13/3
B) 7/3
C) 10/3
D) 4/3
40. Find the coordinates of the point which trisects the line segment joining A(1, 0, -1) and B(4, 3, 2).
A) (2, 1, 0) and (3, 2, 1)
B) (2, 1, 1) and (3, 2, 0)
C) (1, 2, 0) and (2, 3, 1)
D) (2, 0, 1) and (3, 1, 2)
41. In what ratio does the point (2, 3, 4) divide the line segment joining A(1, 2, 3) and B(3, 4, 5)?
A) 1:1
B) 2:1
C) 1:2
D) 3:1
42. The point (x, y, z) divides the line segment joining A(x1, y1, z1) and B(x2, y2, z2) in the ratio k:1. What are the coordinates of the point?
A) ((kx2 + x1)/(k+1), (ky2 + y1)/(k+1), (kz2 + z1)/(k+1))
B) ((x2 + kx1)/(k+1), (y2 + ky1)/(k+1), (z2 + kz1)/(k+1))
C) ((kx2 - x1)/(k-1), (ky2 - y1)/(k-1), (kz2 - z1)/(k-1))
D) ((x1 - kx2)/(1-k), (y1 - ky2)/(1-k), (z1 - kz2)/(1-k))
43. Find the coordinates of the point that divides the line segment joining A(-2, 3, 5) and B(1, -4, 2) externally in the ratio 1:2.
A) (-4, 17, 8)
B) (8, -17, 4)
C) (-4, 8, 17)
D) (17, -4, 8)
44. A point R divides the line segment PQ joining P(2, -1, 3) and Q(4, 5, 1) in the ratio 3:4 internally. Find the coordinates of R.
A) (20/7, 17/7, 15/7)
B) (17/7, 20/7, 15/7)
C) (20/7, 15/7, 17/7)
D) (15/7, 20/7, 17/7)
45. If a point P divides the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 1:1 internally, what are the coordinates of P?
A) (5/2, 7/2, 9/2)
B) (3, 4, 5)
C) (2, 3, 4)
D) (1, 1, 1)
46. What are the coordinates of the midpoint of the line segment joining A(x1, y1, z1) and B(x2, y2, z2)?
A) ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
B) ((x1 - x2)/2, (y1 - y2)/2, (z1 - z2)/2)
C) (x1 + x2, y1 + y2, z1 + z2)
D) (x1 * x2, y1 * y2, z1 * z2)
47. Find the coordinates of the point which divides the line segment joining the points (1, 2, 3) and (4, 5, 6) in the ratio 2:1 externally.
A) (2, 1, 0)
B) (0, 1, 2)
C) (1, 2, 0)
D) (0, 2, 1)
48. Find the coordinates of the point which divides the line segment joining the points (2, 3, 4) and (5, 6, 7) in the ratio 1:2 internally.
A) (11/3, 13/3, 5)
B) (13/3, 11/3, 5)
C) (11/3, 5, 13/3)
D) (5, 11/3, 13/3)
49. If a point P divides the line segment joining A(x1, y1, z1) and B(x2, y2, z2) externally in the ratio m:n, what are the coordinates of P?
A) ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))
B) ((nx2 + mx1)/(m+n), (ny2 + my1)/(m+n), (nz2 + mz1)/(m+n))
C) ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n), (mz2 - nz1)/(m-n))
D) ((nx2 - mx1)/(n-m), (ny2 - my1)/(n-m), (nz2 - mz1)/(n-m))
50. If a point P divides the line segment joining A(x1, y1, z1) and B(x2, y2, z2) internally in the ratio m:n, what are the coordinates of P?
A) ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))
B) ((nx2 + mx1)/(m+n), (ny2 + my1)/(m+n), (nz2 + mz1)/(m+n))
C) ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n), (mz2 - nz1)/(m-n))
D) ((nx2 - mx1)/(n-m), (ny2 - my1)/(n-m), (nz2 - mz1)/(n-m))
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