Online Test — Introduction to Three Dimensional Geometry
10 Questions • 20 min • Chapter MCQ
20:00
Question 1 of 10
easy
A bird physically flies from a stable tree branch located at (2, 3, 5) directly to another branch at (5, 7, 5). The true linear distance covered measures:
5 units
25 units
7 units
√45 units
Explanation: Evaluating linear flight: d = √((5−2)² + (7−3)² + (5−5)²) = √(3² + 4² + 0) = √(9 + 16) = √25 = 5 straight units.
Question 2 of 10
medium
What is the proper ratio in which the xz-plane divides the line segment spanning between (2, −3, 1) and (3, 4, 5)?
3:4 internally
3:4 externally
4:3 internally
4:3 externally
Explanation: The xz-plane is explicitly where y = 0. It divides a segment in the ratio −y₁/y₂. Here, ratio = −(−3)/4 = 3/4. Since the fraction is positive, it signifies an internal division of 3:4.
Question 3 of 10
easy
The locus of a point for which y = 0 and z = 0 is:
xy-plane
yz-plane
z-axis
x-axis
Explanation: When both y and z coordinates are zero, the x-coordinate can be any real number. This is the exact definition of the x-axis.
Question 4 of 10
easy
The midpoint of the line segment joining A(3, 4, 5) and B(1, 2, 3) is:
(2, 3, 4)
(1, 1, 1)
(4, 6, 8)
(2, 2, 2)
Explanation: Using the midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). Thus, ((3+1)/2, (4+2)/2, (5+3)/2) = (4/2, 6/2, 8/2) = (2, 3, 4).
Question 5 of 10
easy
The distance between points (0,0,5) and (0,12,5) is:
5
10
12
13
Explanation: Distance = √(0²+12²+0²)=12.
Question 6 of 10
medium
Which point lies in the sixth octant?
(2,−3,−4)
(−2,3,−4)
(−2,−3,4)
(2,3,−4)
Explanation: Signs (+,−,−) correspond to the sixth octant.
Question 7 of 10
medium
The length of the projection of the line segment joining A(1, 2, 3) and B(4, 6, 8) strictly onto the xy-plane is:
5
25
√34
√41
Explanation: The projection onto the xy-plane ignores the z-coordinates. Length = √((4−1)² + (6−2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.
Question 8 of 10
medium
If the midpoint of A(x,4,6) and B(8,2,10) is (5,3,8), then x =
1
2
3
4
Explanation: (x+8)/2=5 ⇒ x=2.
Question 9 of 10
medium
The perpendicular distance of the point P(a, b, c) from the z-axis is:
√(a² + b²)
√(b² + c²)
√(a² + c²)
|c|
Explanation: The shortest distance from the z-axis ignores the z-coordinate since the perpendicular drops onto the point (0, 0, c). Distance = √((a−0)² + (b−0)² + (c−c)²) = √(a² + b²).
Question 10 of 10
medium
The spatial point (−5, 4, −3) is properly positioned in which octant?
II
IV
VI
VIII
Explanation: The signs are (−, +, −). Since z is negative, it's one of the bottom octants. The xy-coordinates (−, +) form the 2nd quadrant. The 2nd quadrant below the xy-plane translates to Octant VI.