Introduction to Three Dimensional Geometry - Practice Questions with Answers

Q1.

The three coordinate planes divide space into how many octants?

• A 8
• B 12
• C 4
• D 6

Q2.

The coordinates of the origin in three dimensional space are:

• A (1, 0, 0)
• B (0, 0, 0)
• C (1, 1, 1)
• D (0, 0, 1)

Q3.

A point lying in the XY-plane has which coordinate equal to zero?

• A x
• B y
• C z
• D none of them

Q4.

A point lying on the x-axis has coordinates of the form:

• A (0, y, 0)
• B (0, 0, z)
• C (x, y, 0)
• D (x, 0, 0)

Q5.

The distance of the point P(x, y, z) from the origin is:

• A √(x² + y² + z²)
• B x² + y² + z²
• C √(x + y + z)
• D x + y + z

Q6.

The distance between the points (1, 2, 3) and (1, 2, 5) is:

• A 1
• B 2
• C 3
• D 5

Q7.

The YZ-plane is characterised by the equation:

• A z = 0
• B x = y
• C x = 0
• D y = 0

Q8.

The point (2, 3, 4) lies in which octant?

• A Second
• B Fourth
• C Eighth
• D First

Q9.

The midpoint of the segment joining (2, 4, 6) and (4, 8, 10) is:

• A (3, 6, 8)
• B (6, 12, 16)
• C (2, 4, 4)
• D (1, 2, 2)

Q10.

How many coordinate planes are there in three dimensional geometry?

• A 2
• B 3
• C 4
• D 8

Q11.

The ZX-plane is described by the equation:

• A x = z
• B x = 0
• C y = 0
• D z = 0

Q12.

The distance between the origin and the point (3, 4, 12) is:

• A 19
• B √19
• C 12
• D 13

Q13.

A point on the z-axis has coordinates of the form:

• A (0, 0, z)
• B (z, 0, 0)
• C (0, z, 0)
• D (z, z, 0)

Q14.

In the coordinate triple (x, y, z), the value z is measured along the:

• A y-axis
• B z-axis
• C line y = x
• D x-axis

Q15.

The distance between (0, 0, 0) and (1, 1, 1) is:

• A 1
• B 3
• C √3
• D √2

Q16.

The three coordinate axes in space are:

• A Inclined at 60° to one another
• B Coincident at every point
• C Parallel to one another
• D Mutually perpendicular

Q17.

The point (0, 5, 0) lies on the:

• A y-axis
• B z-axis
• C XY-plane only
• D x-axis

Q18.

The section formula for internal division of PQ in ratio m : n gives the x-coordinate as:

• A (x₁ + x₂)/2
• B (mx₂ + nx₁)/(m + n)
• C (mx₂ − nx₁)/(m − n)
• D (mx₁ + nx₂)/(m + n)

Q19.

The midpoint formula is the section formula applied with ratio:

• A 2 : 1
• B 0 : 1
• C 1 : 1
• D 1 : 2

Q20.

The distance between the points (1, 0, 0) and (0, 1, 0) is:

• A 2
• B 0
• C 1
• D √2

Q21.

The distance between the points A(2, 3, 5) and B(4, 3, 1) is:

• A 2√5
• B 4
• C √20 units measured along the y-axis only
• D 6

Q22.

The point that divides the join of (1, −2, 3) and (3, 4, −5) internally in the ratio 1 : 3 is:

• A (1/2, −5/2, 4)
• B (3/2, −1/2, 1)
• C (2, 1, −1)
• D (5/2, 5/2, −3)

Q23.

The centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (7, 8, 9) is:

• A (12, 15, 18)
• B (2, 3, 4)
• C (4, 5, 6)
• D (3, 4, 5)

Q24.

In what ratio does the XY-plane divide the line joining A(1, 2, 3) and B(2, 4, −6)?

• A 1 : 3
• B 2 : 1
• C 3 : 1
• D 1 : 2

Q25.

The point (−3, 1, 2) lies in the octant where the signs of (x, y, z) are:

• A (−, +, +)
• B (+, +, +)
• C (−, −, +)
• D (+, −, −)

Q26.

If the distance between (2, 3, a) and (2, 3, 1) is 5 units, then a can be:

• A 6 only
• B 6 or −4
• C 4 or −6
• D 5 or −5

Q27.

The point equidistant from the three coordinate axes among the following is:

• A (0, 1, 2)
• B (2, 2, 0)
• C (1, 1, 1)
• D (1, 2, 3)

Q28.

The projection (foot of perpendicular) of the point (3, 4, 5) on the XY-plane is:

• A (0, 0, 5)
• B (3, 0, 5)
• C (0, 4, 5)
• D (3, 4, 0)

Q29.

The point which divides the join of (2, 1, 4) and (4, 3, 2) externally in the ratio 1 : 2 is:

• A (0, −1, 6)
• B (0, 1, 6)
• C (6, 5, 0)
• D (3, 2, 3)

Q30.

The distance of the point (1, 2, 3) from the x-axis is:

• A √14
• B √13
• C 1
• D √5

Q31.

If the point (x, 0, 0) is equidistant from (1, 2, 3) and (3, 2, 1), then x equals:

• A 2
• B 3
• C 0
• D 1

Q32.

The points A(1, 2, 3), B(2, 3, 4) and C(3, 4, 5) are:

• A Vertices of an equilateral triangle
• B Vertices of a right triangle
• C Coincident
• D Collinear

Q33.

The distance of the point (4, 3, 0) from the z-axis is:

• A 5
• B 4
• C 3
• D 0

Q34.

If A(3, 2, 0), B(5, 3, 2) and C(−9, 6, −3) are the vertices of a triangle, the midpoint of BC is:

• A (7, 9/2, 1)
• B (−2, 9/2, −1/2)
• C (−2, 9, −1)
• D (−4, 9/2, −1/2)

Q35.

In what ratio does the YZ-plane divide the line joining (−2, 4, 7) and (3, −5, 8)?

• A 1 : 2
• B 2 : 1
• C 2 : 3
• D 3 : 2

Q36.

The number of points in space whose each coordinate is either 1 or −1 is:

• A 6
• B 4
• C 3
• D 8

Q37.

If the midpoint of the segment joining (a, 2, 3) and (5, b, 7) is (3, 4, 5), then a + b equals:

• A 7
• B 9
• C 11
• D 5

Q38.

The perpendicular distance of the point (2, −3, 4) from the XY-plane is:

• A √29
• B 4
• C 2
• D 3

Q39.

The triangle with vertices (0, 0, 0), (3, 0, 0) and (0, 4, 0) is:

• A Obtuse angled
• B Not a valid triangle
• C Right angled at the origin
• D Equilateral

Q40.

If P(2, 3, 4) and Q(4, 5, 6), the point dividing PQ in ratio 3 : 1 internally is:

• A (5/2, 7/2, 9/2)
• B (3, 4, 5)
• C (10/3, 13/3, 16/3)
• D (7/2, 9/2, 11/2)

Q41.

Why can the 2D test for collinearity using slopes not be carried over directly to three dimensions?

• A A line in space has no single slope, since it makes three separate angles with the axes
• B Slopes only exist for lines passing through the origin in any dimension
• C Three points in space can never be collinear in the majority of documented cases
• D Slope is undefined whenever any coordinate is negative under usual circumstances

Q42.

When solving the section formula for an unknown ratio, obtaining a negative value of k means that the point:

• A Lies at the origin of the coordinate system
• B Divides the segment externally rather than internally
• C Does not lie on the line through the two given points
• D Coincides with the midpoint of the segment

Q43.

The locus of a point whose distance from the z-axis is a constant a is:

• A A circle of radius a lying in the XY-plane only
• B A pair of planes parallel to the XY-plane
• C A cylinder of radius a with the z-axis as its axis
• D A sphere of radius a centred at the origin

Q44.

A point equidistant from all four vertices of a tetrahedron is found by:

• A Averaging the four vertices, which always gives the required point
• B Taking the midpoint of the longest edge of the tetrahedron
• C Projecting the centroid onto the XY-plane in every case
• D Solving the three equations obtained by equating squared distances pairwise

Q45.

The points A(1, 2, 3), B(−1, −2, −1), C(2, 3, 2) and D(4, 7, 6) form:

• A A parallelogram, since both pairs of opposite sides are equal and the diagonals bisect each other
• B An equilateral triangle with one repeated vertex according to standard results
• C A regular tetrahedron in most documented configurations
• D Four collinear points under typical conditions

Q46.

If a point P divides AB in the ratio k : 1 and lies on the XY-plane, then k is given by:

• A z₁ + z₂
• B −z₁/z₂
• C z₁/z₂
• D −z₂/z₁

Q47.

The distance formula in three dimensions follows from applying the Pythagorean theorem:

• A Three times, one for each coordinate plane involved
• B Only when all coordinates are positive as generally observed
• C Twice - once in the base plane and once in the vertical direction
• D Once, exactly as in two dimensions with an extra term appended

Q48.

The point on the y-axis equidistant from A(3, 1, 2) and B(5, 5, 2) is:

• A (0, 3, 0)
• B (0, 0, 5)
• C (5, 0, 0)
• D (0, 5, 0)

Q49.

If the origin is the centroid of a triangle with vertices (2a, 2, 6), (−4, 3b, −10) and (8, 14, 2c), then a, b and c are:

• A a = −2, b = −16/3, c = 2
• B a = 2, b = 16/3, c = −2
• C a = −2, b = 16/3, c = −2
• D a = 4, b = −5, c = 1

Q50.

Three points are collinear in space if and only if:

• A They lie in the same octant of the coordinate system
• B The largest of the three pairwise distances equals the sum of the other two
• C All three pairwise distances are equal to one another
• D Their coordinates sum to zero in each of the three components

Q51.

The locus of a point equidistant from the points A and B in space is:

• A A sphere with AB as its diameter in most cases
• B The single midpoint of AB and no other point
• C The plane perpendicular to AB through its midpoint
• D The straight line AB extended indefinitely

Q52.

A point P lies on the x-axis and is at distance 5 from the point (0, 3, 4). The coordinates of P are:

• A (5, 0, 0) or (−5, 0, 0)
• B (3, 0, 0) or (−3, 0, 0)
• C No such point exists
• D (0, 0, 0) only

Q53.

If A(1, 2, 3) and the midpoint of AB is (2, 3, 4), then the coordinates of B are:

• A (3, 4, 5)
• B (1, 1, 1)
• C (4, 6, 8)
• D (3/2, 5/2, 7/2)

Q54.

The ratio in which the line joining (2, 4, 5) and (3, 5, −4) is divided by the XY-plane is:

• A 5 : 1
• B 5 : 4
• C 4 : 5
• D 1 : 1

Q55.

Why does a point's distance from a coordinate axis use only two of its three coordinates?

• A Distances in space are always computed pairwise between two coordinates only
• B The third coordinate is always zero for points measured from an axis
• C The perpendicular from the point meets the axis at the foot sharing that axis coordinate, so that coordinate contributes nothing
• D Every axis lies in the XY-plane, making the third coordinate irrelevant

Q56.

The vertices A(0, 7, 10), B(−1, 6, 6) and C(−4, 9, 6) form a triangle that is:

• A Equilateral
• B Scalene with no right angle
• C Degenerate, since the points are collinear
• D Right angled and isosceles

Q57.

For a point dividing a segment externally, the denominator in the section formula is:

• A m − n, which is why m = n gives no valid point
• B m + n, exactly as in internal division
• C Always 2, independent of the ratio chosen
• D mn, the product of the two ratio terms

Q58.

The point (1, 2, −3) lies in the octant characterised by:

• A (+, +, +)
• B (+, +, −)
• C (+, −, −)
• D (−, +, −)

Q59.

If a parallelepiped is formed with edges along the axes and one vertex at the origin and the opposite vertex at (a, b, c), its main diagonal has length:

• A abc
• B √(ab + bc + ca)
• C √(a² + b² + c²)
• D a + b + c

Q60.

The centroid of a tetrahedron with vertices A, B, C and D is found by:

• A Averaging only three vertices and ignoring the fourth
• B Taking the midpoint of the longest edge in every case
• C Projecting the triangle centroid of ABC onto the fourth vertex
• D Averaging all four position vectors, dividing each coordinate sum by 4

Q61.

The distance between the points (1, 2, 3) and (4, 6, 3) is:

• A 5
• B √29
• C 6
• D 4

Q62.

The midpoint of the segment joining (2, 4, 6) and (4, 8, 10) is:

• A (3, 6, 8)
• B (6, 12, 16)
• C (2, 4, 4)
• D (1, 2, 2)

Q63.

The point dividing the segment from (1, 2, 3) to (4, 5, 6) in the ratio 2:1 internally is:

• A (3, 4, 5)
• B (2, 3, 4)
• C (5, 6, 7)
• D (3, 3, 4)

Q64.

The signs of the coordinates of the point (−1, 2, −3) are:

• A (+, +, +)
• B (−, +, −)
• C (−, −, −)
• D (+, −, +)

Q65.

The centroid of the triangle with vertices (1, 2, 3), (2, 3, 4) and (3, 4, 5) is:

• A (2, 3, 4)
• B (3, 4, 5)
• C (1, 2, 3)
• D (6, 9, 12)

Q66.

The point on the x-axis equidistant from (1, 2, 3) and (3, 2, −1) is:

• A (0, 0, 0)
• B (1, 0, 0)
• C (2, 0, 0)
• D (−1, 0, 0)

Q67.

The distance of the point (3, 4, 5) from the origin is:

• A 5√2
• B 5
• C 10
• D √48

Q68.

The reflection of the point (1, 2, 3) in the xy-plane is:

• A (1, 2, −3)
• B (−1, −2, 3)
• C (1, −2, 3)
• D (−1, 2, 3)