MathsISC Class 11 · Introduction to Three-dimensional Geometry

Section Formula, Midpoint and Centroid in Space and Intersection with Coordinate Planes

Q: What does a negative ratio mean?

The point divides the segment externally, lying outside AB on the line.

Q: Are there three medians in 3D?

Yes, they meet at the centroid, as in a plane triangle.

Q: Can a line meet two coordinate planes?

Yes, it generally meets each of the three coordinate planes once.

Updated 17 October 2026 · Fact-checked

The point dividing the join of A(x₁, y₁, z₁) and B(x₂, y₂, z₂) internally in the ratio m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)); for external division, replace n by −n. The midpoint uses m = n = 1, and the centroid of a triangle is the average of the vertices.

Internal division lies between A and B, the midpoint is the equal case, and external division lies beyond; y and z follow the same pattern as x.

Illustration drawn for QuestionBuddy.

Understand Section Formula, Midpoint and Centroid in Space

Internal division. The point P dividing AB internally in the ratio $m : n$ has coordinates $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$ .

External division. In the ratio $m : n$ externally, $(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n}, \frac{m z _{2} - n z _{1}}{m - n})$ . The point then lies on the line AB outside the segment.

Midpoint. $m = n$ : $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$ .

Centroid. The centroid of triangle ABC is $(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3}, \frac{z _{1} + z _{2} + z _{3}}{3})$ ; for a tetrahedron the average of the four vertices.

Meeting a coordinate plane. To find where the line joining $A$ and $B$ meets the xy-plane (z = 0), let the ratio be $λ : 1$ and set the z-coordinate to 0: $λ = - \frac{z _{1}}{z _{2}}$ . A negative value means external division. The yz-plane uses x and the zx-plane uses y in the same way.

Parallelograms. If ABCD is a parallelogram, the diagonals bisect each other, so the midpoint of AC equals the midpoint of BD; use this to find the fourth vertex: $D = A + C - B$ coordinate by coordinate.

Section Formula, Midpoint and Centroid in Space: formulas to remember

Internal division: $\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n}\right)$; Ratio AP : PB = m : n.
External division: $Replace n by -n: denominator $m - n$$; The point lies outside the segment.
Midpoint and centroid: $M i d p o in t : a v er a g eo f tw o ∣ C e n t r o i d : a v er a g eo f t h r ee v er t i ces$; Triangle in space.
Meeting the xy-plane: $λ = -\frac{z_1}{z_2}$; λ < 0 means external division.

Section Formula, Midpoint and Centroid in Space in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Placing a mid-air relay

Suppose a relay must be placed on the straight line from a tower base A(1, −2, 3) to a mountain top B(4, 7, −9) at a point dividing AB in the ratio 2 : 1.

Find its coordinates.

Try it first, then show the solutionSolution

x: $\frac{2 \times 4 + 1 \times 1}{3} = 3$ .
y: $\frac{2 \times 7 + 1 \times ( - 2 )}{3} = 4$ .
z: $\frac{2 \times ( - 9 ) + 1 \times 3}{3} = - 5$ , so the relay is at (3, 4, −5).

Answer: (3, 4, −5).

Memory hook: The nearer vertex gets the larger multiplier (B is multiplied by m, A by n).

Tricks to remember Section Formula, Midpoint and Centroid in Space

Short memory aids. Say each one aloud once, then use it on a question the same day.

Cross-multiply the ratio
The coordinate of B is multiplied by m, that of A by n.
Use it when: Remembering the section formula.
Check the midpoint
A mistake in the ratio can be checked by testing m = n.
Use it when: Verification.

Section Formula, Midpoint and Centroid in Space: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Midpoint · 1 mark · MCQBoard level

The midpoint of (2, 4, 6) and (−4, 0, 2) is:
(a) $(- 1, 2, 4)$
(b) $(- 2, 4, 8)$
(c) $(1, 2, 4)$
(d) $(- 1, - 2, 4)$

Try it first, then show the solutionSolution

$(\frac{2 - 4}{2}, \frac{4 + 0}{2}, \frac{6 + 2}{2}) = (- 1, 2, 4)$ .

Answer: (a) $(- 1, 2, 4)$

Example 2: Internal division · 3 marks · Short answerBoard level

Find the coordinates of the point dividing the join of A(2, −3, 5) and B(8, 6, −1) internally in the ratio 1 : 2.

Try it first, then show the solutionSolution

$x = \frac{1 ( 8 ) + 2 ( 2 )}{3} = 4$ . (1 mark)
$y = \frac{1 ( 6 ) + 2 ( - 3 )}{3} = 0$ . (1 mark)
$z = \frac{1 ( - 1 ) + 2 ( 5 )}{3} = 3$ ; the point is (4, 0, 3). (1 mark)

Answer: (4, 0, 3)

Example 3: External division · 4 marks · Short answerHarder board level

Find the point which divides the join of A(2, −1, 3) and B(4, 3, 1) externally in the ratio 3 : 2. Verify that A, B and the point are collinear.

Try it first, then show the solutionSolution

$x = \frac{3 ( 4 ) - 2 ( 2 )}{3 - 2} = 8$ ; $y = \frac{3 ( 3 ) - 2 ( - 1 )}{1} = 11$ ; $z = \frac{3 ( 1 ) - 2 ( 3 )}{1} = - 3$ . (2 marks)
The point is P(8, 11, −3). (0.5 mark)
$A B = 4 + 16 + 4 = 26$ , $B P = 16 + 64 + 16 = 46$ , $A P = 36 + 144 + 36 = 66$ . (1 mark)
$A B + B P = A P$ : collinear, with $A P : B P = 3 : 2$ . (0.5 mark)

Answer: P(8, 11, −3); collinear.

Example 4: Centroid and fourth vertex · 4 marks · Short answerHarder board level

Find the centroid of the triangle with vertices A(2, 3, 5), B(4, −1, 7) and C(6, 1, −3). If ABCD is a parallelogram, find D.

Try it first, then show the solutionSolution

Centroid $= (\frac{12}{3}, \frac{3}{3}, \frac{9}{3}) = (4, 1, 3)$ . (1.5 marks)
Diagonals AC and BD bisect each other: midpoint of AC $= (4, 2, 1)$ . (1 mark)
$D = 2 \times (4, 2, 1) - B = (8 - 4, 4 + 1, 2 - 7) = (4, 5, - 5)$ . (1.5 marks)

Answer: G(4, 1, 3); D(4, 5, −5).

Example 5: Meeting a coordinate plane · 5 marks · Long answerHOTS / competency

The line joining P(3, 2, −4) and Q(1, 6, 8) meets the xy-plane at R and the zx-plane at S. Find the ratios in which R and S divide PQ and find R and S.

Try it first, then show the solutionSolution

xy-plane (z = 0): $λ = - \frac{z _{1}}{z _{2}} = \frac{4}{8} = \frac{1}{2}$ , positive, so internal division 1 : 2. (1.5 marks)
R: $x = \frac{1 \cdot 1 + 2 \cdot 3}{3} = \frac{7}{3}$ , $y = \frac{1 \cdot 6 + 2 \cdot 2}{3} = \frac{10}{3}$ ; R $= (\frac{7}{3}, \frac{10}{3}, 0)$ . (1.5 marks)
zx-plane (y = 0): $λ = - \frac{y _{1}}{y _{2}} = - \frac{2}{6} = - \frac{1}{3}$ , negative, so external division 1 : 3. (1 mark)
S: using $t = - \frac{1}{2}$ along PQ: $x = 3 + (- 2) (- \frac{1}{2}) = 4$ , $z = - 4 + 12 (- \frac{1}{2}) = - 10$ ; S $= (4, 0, - 10)$ . (1 mark)

Answer: R(7/3, 10/3, 0), 1 : 2 internally; S(4, 0, −10), 1 : 3 externally.

Where students lose marks in Section Formula, Midpoint and Centroid in Space

The slips examiners see most often, and what to write instead.

Mistake: Swapping m and n
Multiplying the wrong coordinates.
Do this: The coordinates of B are multiplied by m and those of A by n.
Usually costs: 2 marks.
Mistake: Using the internal formula for external division
Forgetting the minus sign.
Do this: External: m − n in the denominator.
Usually costs: 2 marks.
Mistake: Forgetting to divide the centroid by 3
Giving the sum.
Do this: Divide each coordinate sum by 3.
Usually costs: 1 mark.

Exam tips

Name the ratio as AP : PB = m : n before substituting.
Test the midpoint case as a check.
Use diagonal midpoints for parallelograms.
Interpret negative ratios as external division.

Practice questions on Section Formula, Midpoint and Centroid in Space

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The centroid of the triangle with vertices (0, 0, 0), (3, 0, 0) and (0, 6, 3) is:
(a) (1, 2, 1)
(b) (3, 6, 3)
(c) (1, 1, 1)
(d) (2, 2, 2)
Hint
Average each coordinate.
Answer
(a) (1, 2, 1)
Q2 · 2 marks · Short answerBoard level
Find the midpoint of the line joining (3, −2, 5) and (−1, 6, −9).
Hint
Average each coordinate.
Answer
$(1, 2, - 2)$ .
Q3 · 3 marks · Short answerHarder board level
Find the point which divides (3, −1, 7) and (9, 5, −2) internally in the ratio 1 : 2.
Hint
m : n = 1 : 2.
Answer
$x = \frac{9 + 6}{3} = 5$ ; $y = \frac{5 - 2}{3} = 1$ ; $z = \frac{- 2 + 14}{3} = 4$ .
Q4 · 3 marks · Short answerHarder board level
The origin is the centroid of the triangle with vertices (a, 3, 5), (−2, b, −7) and (1, −9, c). Find a, b and c.
Hint
The sum of each coordinate is zero.
Answer
$a - 2 + 1 = 0$ : $a = 1$ ; $3 + b - 9 = 0$ : $b = 6$ ; $5 - 7 + c = 0$ : $c = 2$ .
Q5 · 3 marks · Short answerHarder board level
In what ratio does the yz-plane divide the join of (−2, 4, 7) and (3, −5, 8)?
Hint
Set x = 0.
Answer
$λ = - \frac{x _{1}}{x _{2}} = \frac{2}{3}$ ; internally in the ratio 2 : 3.
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose three drones fly at A(1, 2, 3), B(3, 4, 5) and C(5, 0, 7).
(i) Find the centroid of the formation. (1 mark)
(ii) Find the midpoint of AB. (1 mark)
(iii) Find the point dividing AC in ratio 1 : 3 internally. (1 mark)
(iv) Find the fourth vertex D of the parallelogram ABCD. (1 mark)
Hint
Use the formulas in turn.
Answer
(i) $(3, 2, 5)$ . (ii) $(2, 3, 4)$ . (iii) $(\frac{5 + 3}{4}, \frac{0 + 6}{4}, \frac{7 + 9}{4}) = (2, 1.5, 4)$ . (iv) $D = A + C - B = (3, - 2, 5)$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Midpoint formula?Show answer
Average of the coordinates.
2.Centroid of a triangle?Show answer
Average of three vertices.
3.Ratio for the xy-plane?Show answer
−z₁ : z₂.
4.Fourth vertex of a parallelogram?Show answer
A + C − B.

Section Formula, Midpoint and Centroid in Space: frequently asked questions

What does a negative ratio mean?

The point divides the segment externally, lying outside AB on the line.

Are there three medians in 3D?

Yes, they meet at the centroid, as in a plane triangle.

Can a line meet two coordinate planes?

Yes, it generally meets each of the three coordinate planes once.