MathsICSE Class 9 · Mid-point and Intercept Theorems

The Mid-point Theorem, Its Converse and the Triangle of Mid-points (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. Conversely, a line through the mid-point of one side parallel to a second side bisects the third side. Joining the three mid-points divides the triangle into four congruent triangles.

Joining the mid-points gives EF ∥ BC with EF = ½ BC, and the four equal triangles each have ¼ of the area of ABC.

Illustration drawn for QuestionBuddy.

Understand The Mid-point Theorem, Its Converse and the Triangle of Mid-points

Mid-point theorem (prove). In $△ A B C$ , if $D$ and $E$ are the mid-points of $A B$ and $A C$ , then $D E ∥ B C$ and $D E = \frac{1}{2} B C$ .

*Proof.* Extend $D E$ to $F$ so that $E F = D E$ . Join $C F$ . In $△ A D E$ and $△ C F E$ : $A E = C E$ , $∠ A E D = ∠ C E F$ (vertically opposite), $D E = F E$ , so they are congruent (SAS) and $A D = C F$ , $∠ D A E = ∠ F C E$ . The equal alternate angles give $A D ∥ C F$ , so $B D ∥ C F$ and $B D = A D = C F$ . Therefore $B C F D$ is a parallelogram, giving $D F ∥ B C$ and $D F = B C$ . Since $D E = \frac{1}{2} D F$ , the result follows.

Converse (prove). If a line through the mid-point $D$ of $A B$ is parallel to $B C$ and meets $A C$ at $E$ , then $E$ is the mid-point of $A C$ .

*Proof.* Draw $E F ∥ A B$ meeting $B C$ at $F$ . Then $D E F B$ is a parallelogram, so $D E = B F$ and $D B = E F$ . In $△ A D E$ and $△ E F C$ : $A D = D B = E F$ , $∠ A D E = ∠ E F C$ and $∠ A E D = ∠ E C F$ (corresponding angles), so the triangles are congruent (AAS) and $A E = E C$ .

Triangle of mid-points. If $D, E, F$ are the mid-points of $B C, C A, A B$ , then $E F = \frac{1}{2} B C$ , $F D = \frac{1}{2} C A$ , $D E = \frac{1}{2} A B$ . The four triangles $A F E$ , $F B D$ , $E D C$ and $D E F$ are congruent (SSS), so
$ $perimeter (△ D E F) = \frac{1}{2} perimeter (△ A B C), ar (△ D E F) = \frac{1}{4} ar (△ A B C) .$ $
Also $A F D E$ is a parallelogram.

Using it in algebra. If $B C = 5 x - 4$ and $D E = 2 x + 1$ , then $2 x + 1 = \frac{5 x - 4}{2}$ , so $x = 6$ , $B C = 26$ , $D E = 13$ .

Level. Class IX is a school-exam class. Direct length questions are 2 to 3 marks; the converse proof and algebraic setting are at the hardest school-exam level.

The Mid-point Theorem, Its Converse and the Triangle of Mid-points: formulas to remember

Mid-point theorem: $D E ∥ B C, D E = \frac{1}{2} B C$; D, E mid-points of AB, AC.
Converse: $D E ∥ B C, A D = D B \Rightarrow A E = E C$; Line through a mid-point.
Perimeter of mid-point triangle: $P (△ D E F) = \frac{1}{2} P (△ A B C)$; Each side is half.
Area of mid-point triangle: $ar (△ D E F) = \frac{1}{4} ar (△ A B C)$; Four congruent triangles.

How to solve The Mid-point Theorem, Its Converse and the Triangle of Mid-points questions

Identify the two mid-points, then name the parallel and the half.

1Mark equal tick marks on the halves of each side.
2Identify the segment joining two mid-points; it is parallel to and half of the third side.
3Write equal lengths as equations if variables appear and solve.
4For the mid-point triangle, double (or halve) all three sides to compare perimeters.
5For proofs, cite the theorem by name and the congruence rule you use.

The Mid-point Theorem, Its Converse and the Triangle of Mid-points 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.

A fence across a triangular plot

Suppose a triangular garden has sides $A B = 30$ m, $B C = 44$ m, $C A = 26$ m, and posts $D$ , $E$ at the mid-points of $A B$ and $A C$ with a wire stretched between them.

Find the length of the wire and the perimeter of $△ A D E$ .

Try it first, then show the solutionSolution

$D E = \frac{1}{2} B C = 22$ m.
$A D = 15$ , $A E = 13$ , so perimeter of $△ A D E = 15 + 13 + 22 = 50$ m, half of 100 m.

Answer: Wire 22 m; perimeter of the small triangle 50 m.

Memory hook: Mid-point segment = half the base, parallel to it.

Shading a quarter

Imagine a triangular sign of area 96 cm² with its mid-point triangle painted red.

What is the red area, and the area of the rest?

Try it first, then show the solutionSolution

Red $= \frac{96}{4} = 24$ cm².
Rest $= 96 - 24 = 72$ cm².

Answer: 24 cm² and 72 cm².

Memory hook: The mid-point triangle is one of four equal parts.

Tricks to remember The Mid-point Theorem, Its Converse and the Triangle of Mid-points

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

Half and parallel
Mid-point segment: parallel to the third side, half its length.
Use it when: Direct questions.
Four equal triangles
Joining mid-points cuts the triangle into four congruent ones.
Use it when: Area and perimeter questions.
Through a mid-point, parallel
It bisects the next side too (converse).
Use it when: Chain of mid-points.

The Mid-point Theorem, Its Converse and the Triangle of Mid-points: solved examples

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

Example 1: Direct length · 1 mark · MCQBoard level

$D$ and $E$ are the mid-points of $A B$ and $A C$ in $△ A B C$ . If $B C = 14$ cm, then $D E$ is:
(a) 5 cm
(b) 7 cm
(c) 14 cm
(d) 28 cm

Try it first, then show the solutionSolution

$D E = \frac{1}{2} B C = 7$ cm.

Answer: (b) 7 cm

Example 2: Triangle of mid-points · 3 marks · Short answerBoard level

In $△ A B C$ , $A B = 22$ cm, $B C = 30$ cm and $C A = 26$ cm. $D, E, F$ are the mid-points of $B C, C A, A B$ . Find the perimeter of $△ D E F$ .

Try it first, then show the solutionSolution

$E F = 15$ , $F D = 13$ , $D E = 11$ .
Perimeter $= 15 + 13 + 11 = 39$ cm (half of 78 cm).

Answer: 39 cm

Example 3: Algebra with the theorem · 4 marks · Short answerHarder board level

$D$ and $E$ are the mid-points of $A B$ and $A C$ . If $B C = (5 x - 4)$ cm and $D E = (2 x + 1)$ cm, find $x$ , $B C$ and $D E$ . If $∠ A D E = 6 2^{\circ}$ , find $∠ B$ .

Try it first, then show the solutionSolution

$D E = \frac{1}{2} B C$ : $2 x + 1 = \frac{5 x - 4}{2}$ .
$4 x + 2 = 5 x - 4$ , so $x = 6$ .
$B C = 26$ cm, $D E = 13$ cm.
$D E ∥ B C$ , so $∠ B = ∠ A D E = 6 2^{\circ}$ (corresponding angles).

Answer: $x = 6$ , $B C = 26$ cm, $D E = 13$ cm, $∠ B = 6 2^{\circ}$

Example 4: AFDE is a parallelogram · 4 marks · Long answerHarder board level

$D, E, F$ are the mid-points of $B C, C A, A B$ of $△ A B C$ . Prove that $A F D E$ is a parallelogram. If $ar (△ A B C) = 96$ cm², find $ar (A F D E)$ and $ar (△ D E F)$ .

Try it first, then show the solutionSolution

$D E ∥ A B$ and $D F ∥ A C$ (mid-point theorem), so $A F D E$ has both pairs of opposite sides parallel: a parallelogram.
Diagonal $E F$ splits it into two congruent triangles, and the four small triangles are congruent, each $\frac{96}{4} = 24$ cm².
$ar (A F D E) = 2 \times 24 = 48$ cm².
$ar (△ D E F) = 24$ cm².

Answer: Proved; ar(AFDE) = 48 cm²; ar(DEF) = 24 cm².

Example 5: Converse with numbers · 5 marks · Long answerHOTS / competency

In $△ A B C$ , $D$ is the mid-point of $A B$ . A line through $D$ parallel to $B C$ meets $A C$ at $E$ , and $F$ is the mid-point of $B C$ . Prove that $E$ is the mid-point of $A C$ . If $A B = 36$ cm and $B C = 42$ cm, find $E F$ and $D E$ .

Try it first, then show the solutionSolution

Draw $E F^{'} ∥ A B$ meeting $B C$ at $F^{'}$ . Then $D E F^{'} B$ is a parallelogram, so $D E = B F^{'}$ and $E F^{'} = D B = A D$ .
In $△ A D E$ and $△ E F^{'} C$ : $A D = E F^{'}$ , $∠ A D E = ∠ E F^{'} C$ and $∠ A E D = ∠ E C F^{'}$ (corresponding angles). By AAS the triangles are congruent, so $A E = E C$ .
Hence $E$ is the mid-point of $A C$ , and then $F^{'}$ is the mid-point of $B C$ , so $F^{'} = F$ .
$E F = \frac{1}{2} A B = 18$ cm and $D E = \frac{1}{2} B C = 21$ cm.

Answer: Proved; EF = 18 cm, DE = 21 cm.

Where students lose marks in The Mid-point Theorem, Its Converse and the Triangle of Mid-points

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

Mistake: Doubling instead of halving
The mid-segment is half the third side.
Wrong working
$B C = 14 \Rightarrow D E = 28$ .
Right working
$D E = 7$ .
Do this: Mid-segment is the shorter one.
Usually costs: Whole answer.
Mistake: Forgetting parallelism
Angle questions need DE ∥ BC.
Wrong working
$∠ A D E = 6 2^{\circ} \Rightarrow ∠ B = 11 8^{\circ}$ .
Right working
$∠ B = 6 2^{\circ}$ (corresponding).
Do this: State the parallel lines.
Usually costs: Reason mark.
Mistake: Taking any line as a mid-segment
Both ends must be mid-points.
Wrong working
$D$ mid-point of $A B$ , $E$ any point of $A C$ , so $D E = \frac{1}{2} B C$ .
Right working
Only if E is the mid-point of AC.
Do this: Check both tick marks.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; the mid-point theorem proof is a standard 4 mark theorem.
Mark equal sides with tick marks before writing any equality.
In the converse, state which angle pair is corresponding.
Mention "mid-point theorem" as the reason for each parallel or half.

Practice questions on The Mid-point Theorem, Its Converse and the Triangle of Mid-points

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 line joining the mid-points of two sides of a triangle is:
(a) parallel to the third side and half of it
(b) perpendicular to the third side
(c) equal to the third side
(d) twice the third side
Hint
Mid-point theorem.
Answer
(a)
Q2 · 3 marks · Short answerHarder board level
$M$ is the mid-point of $B C$ in $△ A B C$ . A line through $M$ parallel to $A B$ meets $A C$ at $N$ . If $A B = 18$ cm and $A C = 26$ cm, find $M N$ and $A N$ .
Hint
Converse: N is the mid-point of AC.
Answer
$N$ is the mid-point of $A C$ , so $A N = 13$ cm; $M N = \frac{1}{2} A B = 9$ cm.
Q3 · 3 marks · Short answerHarder board level
$D, E, F$ are the mid-points of the sides of $△ A B C$ with $A B = 14$ cm and $A C = 10$ cm. Find the perimeter of the parallelogram $A F D E$ .
Hint
AF = 7, AE = 5.
Answer
$A F = 7$ , $A E = 5$ , $F D = A E = 5$ , $D E = A F = 7$ ; perimeter $= 24$ cm.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $D, E$ are the mid-points of $A B, A C$ and $B C = 20$ cm, then $D E = 10$ cm.
Reason (R): The segment joining mid-points of two sides is half the third side.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
Mid-point theorem.
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A triangular garden $A B C$ has $A B = 30$ m, $B C = 44$ m and $C A = 26$ m. Posts $D$ and $E$ are at the mid-points of $A B$ and $A C$ and a wire joins them.
(a) Find the length of the wire $D E$ .
(b) Find the perimeter of $△ A D E$ .
(c) Find the ratio $ar (△ A D E) : ar (△ A B C)$ .
Hint
Each side of ADE is half the corresponding side.
Answer
(a) 22 m (b) 15 + 13 + 22 = 50 m (c) 1 : 4

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.State the mid-point theorem.Show answer
The segment joining the mid-points of two sides is parallel to the third side and half of it.
2.State its converse.Show answer
A line through the mid-point of one side parallel to another side bisects the third side.
3.Ratio of areas of the mid-point triangle to the triangle?Show answer
1 : 4.
4.What quadrilateral is AFDE?Show answer
A parallelogram.

The Mid-point Theorem, Its Converse and the Triangle of Mid-points: frequently asked questions

Do I have to prove the converse?

Both the theorem and its converse are proved in ICSE Class IX.

Does the theorem need a right angle?

No; it works for any triangle.

Why are the four triangles congruent?

Their sides are halves of the sides of the original (SSS).