MathsICSE Class 10 · Section and Mid-point Formula

Mid-point Formula: End Points, Parallelograms and Solved Questions (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

The mid-point of the join of A(x₁, y₁) and B(x₂, y₂) is M((x₁ + x₂)/2, (y₁ + y₂)/2): average the x-coordinates and average the y-coordinates. Turned round, the other end is B = (2x_M − x₁, 2y_M − y₁). Because diagonals of a parallelogram bisect each other, it also finds a missing vertex.

Going from A to M takes the same step (+3, +2) as M to B, so each coordinate of M is the average of the ends.

Illustration drawn for QuestionBuddy.

Understand Mid-point Formula

The mid-point formula is the section formula with ratio $1 : 1$ :

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) .$

In words: each coordinate of the mid-point is the average of the matching coordinates of the ends.

Finding an end point. If $M$ and $A$ are known, $x_{1} + x_{2} = 2 x_{M}$ gives $x_{2} = 2 x_{M} - x_{1}$ , and similarly for $y$ . Think of it as stepping from $A$ to $M$ , then taking the same step again.

Unknowns in coordinates. If the ends contain letters, set each average equal to the matching coordinate of $M$ and solve the two equations separately.

Parallelograms. The diagonals of a parallelogram bisect each other, so for parallelogram $A B C D$ (vertices in order) the mid-points of $A C$ and $B D$ coincide:

$A + C = B + D$ (coordinate by coordinate), so $D = A + C - B$ .

This gives the fourth vertex, and it is also a test: if the mid-points of the diagonals of a quadrilateral are the same, it is a parallelogram. Equal sides then make it a rhombus; equal diagonals make it a rectangle; both make it a square.

Triangles and medians. A median joins a vertex to the mid-point of the opposite side; find that mid-point first, then use the distance formula for the median's length. If the mid-points $D$ , $E$ , $F$ of $B C$ , $C A$ , $A B$ are given, each vertex is the sum of the two mid-points next to it minus the opposite one: $A = E + F - D$ . The segment joining the mid-points of two sides is parallel to the third side and half its length (the mid-point theorem), which gives a quick check.

Where it appears. The mid-point is often a step in a longer question: finding the centre of a circle from a diameter, the centroid via a median, or a perpendicular bisector in the equation-of-a-line chapter.

Mid-point Formula: formulas to remember

Mid-point: $M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$; Average the coordinates.
Other end: $B = (2 x_{M} - x_{1}, 2 y_{M} - y_{1})$; When $M$ and $A$ are known.
Fourth vertex of parallelogram ABCD: $D = (x_{A} + x_{C} - x_{B}, y_{A} + y_{C} - y_{B})$; Diagonals bisect each other.
Vertex from mid-points of sides: $A = E + F - D$; $D$ , $E$ , $F$ are the mid-points of $B C$ , $C A$ , $A B$ .

How to solve Mid-point Formula questions

Average to find a mid-point; double and subtract to find an end.

1Write the coordinates of the known points clearly, with signs.
2For a mid-point, add the x-coordinates and halve; add the y-coordinates and halve.
3For a missing end, use $x_{2} = 2 x_{M} - x_{1}$ and $y_{2} = 2 y_{M} - y_{1}$ .
4For a parallelogram, use 'mid-point of $A C$ = mid-point of $B D$ ' and keep the vertices in order.
5For medians, find the mid-point of the opposite side first.
6Check by averaging your answer with the known end.

Mid-point Formula 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.

Mid-point as a fair meeting place

Suppose two towns sit at $(- 4, 3)$ and $(6, - 1)$ on a map grid (in km), and a hospital is to be built exactly halfway along the straight road between them.

Where should the hospital go?

Try it first, then show the solutionSolution

Average the x-coordinates: $x = \frac{- 4 + 6}{2} = 1$ .
Average the y-coordinates: $y = \frac{3 + ( - 1 )}{2} = 1$ .

Answer: At $(1, 1)$ .

Memory hook: Halfway = the average of each coordinate.

Finding an end from the mid-point

Imagine a footbridge on a site plan whose central pillar is at $(2, 1)$ and whose one end rests at $A (- 3, - 2)$ (in metres).

Where is the other end?

Try it first, then show the solutionSolution

From $A$ to the pillar: $+ 5$ in $x$ , $+ 3$ in $y$ .
Take the same step again: $(2 + 5, 1 + 3) = (7, 4)$ . By formula: $(2 \times 2 - (- 3), 2 \times 1 - (- 2)) = (7, 4)$ .

Answer: At $(7, 4)$ .

Memory hook: Same step again past the middle.

Tricks to remember Mid-point Formula

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

Middle = mean
The mid-point is the mean of the two ends, one coordinate at a time.
Use it when: Every mid-point question.
Double the middle, take away the end
Other end $= 2 M - A$ .
Use it when: Missing end points.
Opposite corners add the same
In a parallelogram $A + C = B + D$ .
Use it when: The fourth vertex and parallelogram tests.

Mid-point Formula: solved examples

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

Example 1: A direct mid-point · 1 mark · MCQBoard level

The mid-point of the join of $(- 3, 8)$ and $(5, - 2)$ is:
(a) $(4, 5)$
(b) $(1, 3)$
(c) $(- 1, 5)$
(d) $(2, 6)$

Try it first, then show the solutionSolution

$(\frac{- 3 + 5}{2}, \frac{8 - 2}{2}) = (1, 3)$ . Option (a) halves the differences instead of the sums.

Answer: (b) $(1, 3)$

Example 2: The other end · 3 marks · Short answerBoard level

$M (2, - 1)$ is the mid-point of $A B$ , where $A$ is $(- 3, 4)$ . Find $B$ .

Try it first, then show the solutionSolution

Let $B = (p, q)$ . $\frac{- 3 + p}{2} = 2 \Rightarrow p = 7$ .
$\frac{4 + q}{2} = - 1 \Rightarrow q = - 6$ .

Answer: $B (7, - 6)$

Example 3: Unknowns in the end points · 3 marks · Short answerHarder board level

The mid-point of the join of $A (2 a - 1, 3)$ and $B (5, b + 1)$ is $(4, - 1)$ . Find $a$ and $b$ .

Try it first, then show the solutionSolution

$\frac{2 a - 1 + 5}{2} = 4 \Rightarrow 2 a + 4 = 8 \Rightarrow a = 2$ .
$\frac{3 + b + 1}{2} = - 1 \Rightarrow b + 4 = - 2 \Rightarrow b = - 6$ .

Answer: $a = 2$ , $b = - 6$

Example 4: Fourth vertex of a parallelogram · 4 marks · Short answerHarder board level

$A (- 1, 2)$ , $B (3, - 1)$ and $C (6, 3)$ are three consecutive vertices of parallelogram $A B C D$ . Find $D$ and the point where the diagonals meet.

Try it first, then show the solutionSolution

Diagonals bisect each other: mid-point of $A C$ = mid-point of $B D$ .
Mid-point of $A C$ $= (\frac{5}{2}, \frac{5}{2})$ .
$D = A + C - B = (- 1 + 6 - 3, 2 + 3 - (- 1)) = (2, 6)$ .
Check: mid-point of $B D = (\frac{5}{2}, \frac{5}{2})$ ✓.

Answer: $D (2, 6)$ ; diagonals meet at $(2.5, 2.5)$ .

Example 5: A triangle from the mid-points of its sides · 4 marks · Short answerHOTS / competency

The mid-points of the sides $B C$ , $C A$ and $A B$ of triangle $A B C$ are $D (1, 2)$ , $E (4, - 1)$ and $F (- 2, 3)$ . Find $A$ , $B$ and $C$ .

Try it first, then show the solutionSolution

$A F D E$ is a parallelogram (mid-point theorem), so $A = E + F - D = (4 - 2 - 1, - 1 + 3 - 2) = (1, 0)$ .
Similarly $B = D + F - E = (1 - 2 - 4, 2 + 3 + 1) = (- 5, 6)$ and $C = D + E - F = (1 + 4 + 2, 2 - 1 - 3) = (7, - 2)$ .
Check: mid-point of $B C = (\frac{- 5 + 7}{2}, \frac{6 - 2}{2}) = (1, 2) = D$ ✓.

Answer: $A (1, 0)$ , $B (- 5, 6)$ , $C (7, - 2)$

Example 6: Case study: a garden plot · 5 marks · Case-basedHOTS / competency

Three corners of a parallelogram-shaped garden plot $A B C D$ are $A (1, 1)$ , $B (5, - 2)$ and $C (8, 2)$ (units of 1 m).
(a) A sprinkler is placed where the diagonals cross. Where is it?
(b) Find corner $D$ .
(c) Show that the plot is a square and find its area.

Try it first, then show the solutionSolution

(a) Mid-point of $A C$ : $(\frac{1 + 8}{2}, \frac{1 + 2}{2}) = (4.5, 1.5)$ .
(b) $D = A + C - B = (1 + 8 - 5, 1 + 2 - (- 2)) = (4, 5)$ . Check: mid-point of $B D = (4.5, 1.5)$ ✓.
(c) $A B = 4^{2} + 3^{2} = 5$ and $B C = 3^{2} + 4^{2} = 5$ , so adjacent sides are equal (a rhombus). Diagonals $A C = 49 + 1 = 50$ and $B D = 1 + 49 = 50$ are equal, so it is a square.
Area $= 5^{2} = 25$ m².

Answer: (a) $(4.5, 1.5)$ (b) $(4, 5)$ (c) Square; 25 m²

Where students lose marks in Mid-point Formula

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

Mistake: Halving the difference instead of the sum
The mid-point averages; it does not measure half the gap.
Wrong working
Mid-point of $(- 3, 8)$ and $(5, - 2)$ : $(\frac{5 - ( - 3 )}{2}, \frac{8 - ( - 2 )}{2}) = (4, 5)$
Right working
$(\frac{- 3 + 5}{2}, \frac{8 + ( - 2 )}{2}) = (1, 3)$
Do this: Say 'add and halve'.
Usually costs: The mark.
Mistake: Taking the mid-point as the other end
The unknown end is beyond the mid-point, not at it.
Wrong working
$B = (\frac{2 + ( - 3 )}{2}, \dots)$
Right working
$B = 2 M - A$
Do this: Write the mid-point equation with the unknown end in it and solve.
Usually costs: Accuracy marks.
Mistake: Using the wrong diagonal pair in a parallelogram
For $A B C D$ in order, the diagonals are $A C$ and $B D$ .
Wrong working
Mid-point of $A B$ = mid-point of $C D$ .
Right working
Mid-point of $A C$ = mid-point of $B D$ .
Do this: Sketch the vertices in order before writing equations.
Usually costs: The whole question.

Exam tips

Write the mid-point as a pair, $(x, y)$ , not as two separate statements; the 2024 report noted a centroid written as ' $x = 0$ , $y = 0$ ' instead of the point $(0, 0)$ , and the same care applies to mid-points.
When naming a quadrilateral, justify each property (diagonals bisect, sides equal, diagonals equal).
Mid-points feed later parts (centroid, median length, equation of a line), so check them by averaging back.
Keep fractions exact, such as $(\frac{5}{2}, \frac{5}{2})$ , unless decimals are clearly fine.

Practice questions on Mid-point Formula

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 mid-point of the join of $(0, 6)$ and $(8, 0)$ is:
(a) $(8, 6)$
(b) $(4, 3)$
(c) $(3, 4)$
(d) $(- 4, 3)$
Hint
Average each coordinate.
Answer
(b) $(4, 3)$
Q2 · 2 marks · Short answerBoard level
The mid-point of a segment is $(- 2, 5)$ and one end is $(3, 1)$ . Find the other end.
Hint
$2 M - A$ .
Answer
$(- 7, 9)$
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $(3, 2)$ is the mid-point of the join of $(1, - 1)$ and $(5, 4)$ .
Reason (R): The mid-point of a segment is found by averaging the coordinates of its ends.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$\frac{- 1 + 4}{2} = 1.5$ .
Answer
(b). The mid-point is $(3, 1.5)$ .
Q4 · 2 marks · Short answerHarder board level
$P (- 1, 2)$ is the mid-point of the join of $A (3, k)$ and $B (m, 5)$ . Find $m$ and $k$ .
Hint
Two separate equations.
Answer
$m = - 5$ , $k = - 1$
Q5 · 3 marks · Short answerHarder board level
Show that $A (1, 1)$ , $B (5, 3)$ , $C (7, 7)$ and $D (3, 5)$ are the vertices of a parallelogram.
Hint
Compare the mid-points of $A C$ and $B D$ .
Answer
Both mid-points are $(4, 4)$ ; the diagonals bisect each other, so $A B C D$ is a parallelogram.
Q6 · 3 marks · Short answerHarder board level
Find the length of the median $A D$ of the triangle with vertices $A (2, 7)$ , $B (- 1, 1)$ and $C (5, - 1)$ .
Hint
$D$ is the mid-point of $B C$ .
Answer
$D (2, 0)$ ; $A D = 7$ units.
Q7 · 4 marks · Short answerHOTS / competency
In triangle $A B C$ , $A$ is $(4, 6)$ ; the mid-point of $A B$ is $D (1, 2)$ and the mid-point of $A C$ is $E (5, 0)$ . Find $B$ and $C$ , and verify that $D E = \frac{1}{2} B C$ .
Hint
$B = 2 D - A$ , $C = 2 E - A$ .
Answer
$B (- 2, - 2)$ , $C (6, - 6)$ . $D E = 16 + 4 = 20$ and $B C = 64 + 16 = 80 = 220$ ✓.

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.Write the mid-point formula.Show answer
$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
2.Mid-point of $(2, 4)$ and $(6, 8)$ ?Show answer
$(4, 6)$
3.Mid-point $(0, 0)$ , one end $(3, - 5)$ . Other end?Show answer
$(- 3, 5)$
4.In parallelogram $A B C D$ , which mid-points coincide?Show answer
Those of the diagonals $A C$ and $B D$ .
5.What is a median of a triangle?Show answer
The segment from a vertex to the mid-point of the opposite side.

Mid-point Formula: frequently asked questions

What is the mid-point formula?

The mid-point of the join of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2): the average of the x-coordinates and the average of the y-coordinates.

How do I find the other end of a segment from its mid-point?

Use (2x_M − x₁, 2y_M − y₁): double the mid-point and subtract the known end.

How is the mid-point formula used for parallelograms?

The diagonals of a parallelogram bisect each other, so the mid-points of AC and BD are the same. This gives the fourth vertex D = A + C − B, and tests whether a quadrilateral is a parallelogram.