MathsICSE Class 9 · Mid-point and Intercept Theorems

Equal Intercepts, Trapezium Mid-line, Quadrilateral of Mid-points and Median to the Hypotenuse (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

Parallel lines that cut equal intercepts on one transversal cut equal intercepts on every transversal. The line joining the mid-points of the non-parallel sides of a trapezium is parallel to the parallel sides and equals half their sum. Joining the mid-points of any quadrilateral forms a parallelogram, and the median to the hypotenuse of a right triangle is half the hypotenuse.

Joining the mid-points of any quadrilateral gives a parallelogram whose sides are parallel to the diagonals and half as long (PQ = ½AC, QR = ½BD).

Illustration drawn for QuestionBuddy.

Understand Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse

Equal intercept theorem. If three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on every transversal. It follows from the converse of the mid-point theorem. Use: a line through the mid-point of one leg of a trapezium, parallel to the bases, bisects the other leg.

Trapezium mid-line. In trapezium $A B C D$ with $A B ∥ D C$ , if $E, F$ are the mid-points of $A D, B C$ , then
$ $E F ∥ A B ∥ D C, E F = \frac{1}{2} (A B + D C) .$ $
The diagonal $A C$ cuts $E F$ at $G$ with $E G = \frac{1}{2} D C$ and $GF = \frac{1}{2} A B$ .

Mid-points of a quadrilateral. If $P, Q, R, S$ are the mid-points of the sides of any quadrilateral $A B C D$ , then $P QR S$ is a parallelogram. Proof: in $△ A B C$ , $P Q ∥ A C$ and $P Q = \frac{1}{2} A C$ ; in $△ A D C$ , $S R ∥ A C$ and $S R = \frac{1}{2} A C$ . So $P Q ∥ S R$ and $P Q = S R$ . The sides of $P QR S$ are half the diagonals: $P Q = \frac{1}{2} A C$ and $QR = \frac{1}{2} B D$ .

Diagonals equal ( $A C = B D$ ): $P QR S$ is a rhombus. (So the mid-points of a rectangle or an isosceles trapezium form a rhombus.)
Diagonals perpendicular: $P QR S$ is a rectangle; area $= \frac{1}{2} A C \times \frac{1}{2} B D$ .
Both: a square.

Median to the hypotenuse. In right-angled $△ A B C$ (right angle at $B$ ) with $M$ the mid-point of $A C$ , $B M = \frac{1}{2} A C$ . Proof: extend $B M$ to $D$ with $M D = B M$ ; $A B C D$ is a parallelogram with a right angle, i.e. a rectangle, whose equal diagonals give $B D = A C$ , so $B M = \frac{1}{2} A C$ . The mid-point of the hypotenuse is therefore equidistant from all three vertices.

Level. Class IX is a school-exam class. The quadrilateral of mid-points and median-to-hypotenuse proofs are 4 to 5 mark questions at the hardest school-exam level.

Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse: formulas to remember

Trapezium mid-line: $E F = \frac{1}{2} (A B + D C)$; E, F mid-points of the legs.
Mid-point quadrilateral: $P Q = \frac{1}{2} A C, QR = \frac{1}{2} B D$; Sides are half the diagonals.
Median to hypotenuse: $B M = \frac{1}{2} A C$; Right angle at B.
Equal diagonals: $A C = B D \Rightarrow P QR S is a rhombus$; Mid-points of rectangle.
Perpendicular diagonals: $A C ⊥ B D \Rightarrow P QR S is a rectangle$; Mid-points of a rhombus.

How to solve Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse questions

Name the mid-points, identify the triangle in which the mid-point theorem applies, and read off the half.

1Mark the mid-points on a clean figure.
2For a trapezium, use the mid-line or apply the theorem in the triangles cut by a diagonal.
3For a quadrilateral, join a diagonal and apply the theorem in both triangles it makes.
4For a right triangle, mark the mid-point of the hypotenuse and use equal distances.
5State the shape from the properties of equal and perpendicular diagonals.

Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse 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 ladder sliding down a wall

Suppose a 13 m ladder leans against a wall with its foot 5 m from the wall; a lantern hangs at the mid-point of the ladder.

How far is the lantern from the corner where wall and floor meet? What if the foot slides out to 12 m?

Try it first, then show the solutionSolution

The wall, floor and ladder form a right triangle with hypotenuse 13 m; the lantern is at the mid-point of the hypotenuse, so its distance from the corner is $\frac{13}{2} = 6.5$ m.
Foot at 12 m: the height is $1 3^{2} - 1 2^{2} = 5$ m, still a right triangle with hypotenuse 13 m, so the distance is again 6.5 m.

Answer: 6.5 m in both positions.

Memory hook: The mid-point of the hypotenuse stays 6.5 m from the right-angle corner.

A trapezoid shelf support

Imagine a trapezium-shaped shelf with parallel edges 30 cm and 12 cm and a support rod joining the mid-points of the slanting sides.

How long is the rod?

Try it first, then show the solutionSolution

$E F = \frac{1}{2} (30 + 12) = 21$ cm.

Answer: 21 cm.

Memory hook: Average of the parallel sides.

Tricks to remember Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse

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

Average of the bases
Trapezium mid-line is the mean of the two parallel sides.
Use it when: EF length.
Sides are half diagonals
Mid-point quadrilateral has sides half the diagonals of the original.
Use it when: Perimeter questions.
Circle on the hypotenuse
The mid-point of the hypotenuse is the centre of the circle through all three vertices.
Use it when: Median questions.

Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse: solved examples

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

Example 1: Equal intercepts · 1 mark · MCQBoard level

Three parallel lines make equal intercepts $A B = B C$ on one transversal. On another transversal they cut $P R = 17$ cm in points $P, Q, R$ in the same order. Then $P Q$ is:
(a) 6 cm
(b) 8 cm
(c) 8.5 cm
(d) 17 cm

Try it first, then show the solutionSolution

Equal intercepts on one transversal give equal intercepts on all, so $P Q = QR = \frac{17}{2} = 8.5$ cm.

Answer: (c) 8.5 cm

Example 2: Median to the hypotenuse · 3 marks · Short answerBoard level

The two shorter sides of a right-angled triangle measure $9$ cm and $12$ cm. Find the length of the median drawn to the longest side.

Try it first, then show the solutionSolution

Longest side (hypotenuse) $= 81 + 144 = 15$ cm.
Median to the hypotenuse $= \frac{1}{2} \times 15 = 7.5$ cm.

Answer: 7.5 cm

Example 3: Trapezium mid-line · 4 marks · Short answerHarder board level

In trapezium $P QR S$ , $P Q ∥ S R$ , $P Q = 26$ cm and $S R = 14$ cm. $E, F$ are the mid-points of $P S$ and $QR$ , and $P R$ meets $E F$ at $G$ . Find $E F$ , $E G$ and $GF$ .

Try it first, then show the solutionSolution

$E F = \frac{1}{2} (26 + 14) = 20$ cm.
In $△ P S R$ , $E$ is the mid-point of $P S$ and $E G ∥ S R$ , so $G$ is the mid-point of $P R$ and $E G = \frac{1}{2} S R = 7$ cm.
In $△ P QR$ , $GF = \frac{1}{2} P Q = 13$ cm.
Check: $7 + 13 = 20$ . ✓

Answer: EF = 20 cm, EG = 7 cm, GF = 13 cm

Example 4: Bisecting the leg · 4 marks · Short answerHarder board level

In trapezium $A B C D$ , $A B ∥ D C$ , $E$ is the mid-point of $A D$ and a line through $E$ parallel to $A B$ meets $B C$ at $F$ . If $A B = 30$ cm, $D C = 18$ cm and $B C = 26$ cm, find $E F$ and $B F$ .

Try it first, then show the solutionSolution

By the equal intercept theorem (or the converse of the mid-point theorem), $F$ is the mid-point of $B C$ .
$E F = \frac{1}{2} (30 + 18) = 24$ cm.
$B F = \frac{1}{2} \times 26 = 13$ cm.

Answer: EF = 24 cm; BF = 13 cm

Example 5: Mid-point quadrilateral · 5 marks · Long answerHOTS / competency

$P, Q, R, S$ are the mid-points of the sides $A B, B C, C D, D A$ of quadrilateral $A B C D$ with diagonals $A C = 10$ cm and $B D = 14$ cm. Prove that $P QR S$ is a parallelogram, find its perimeter, and its area if $A C ⊥ B D$ .

Try it first, then show the solutionSolution

In $△ A B C$ : $P Q ∥ A C$ , $P Q = \frac{1}{2} A C = 5$ . In $△ A D C$ : $S R ∥ A C$ , $S R = 5$ .
So $P Q ∥ S R$ and $P Q = S R$ : $P QR S$ is a parallelogram. Similarly $QR = S P = \frac{1}{2} B D = 7$ .
Perimeter $= 2 (5 + 7) = 24$ cm.
If $A C ⊥ B D$ , then $P Q ⊥ QR$ (as $P Q ∥ A C$ and $QR ∥ B D$ ), so $P QR S$ is a rectangle of area $5 \times 7 = 35$ cm².

Answer: Proved; perimeter 24 cm; area 35 cm² when the diagonals are perpendicular.

Where students lose marks in Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse

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

Mistake: Using the full sum for the mid-line
The mid-line is half the sum.
Wrong working
$E F = 18 + 10 = 28$ .
Right working
$E F = \frac{1}{2} (28) = 14$ .
Do this: Average of the two bases.
Usually costs: Whole answer.
Mistake: Wrong shape for the mid-point quadrilateral
The shape depends on the diagonals of the original figure.
Wrong working
Mid-points of a rectangle form a rectangle.
Right working
A rhombus, since the diagonals are equal.
Do this: Compare the diagonals.
Usually costs: Reason mark.
Mistake: Median to a leg
The result is for the hypotenuse only.
Wrong working
Median to a leg $= \frac{1}{2}$ of that leg.
Right working
Only the median to the hypotenuse is half the side.
Do this: Check which side the median goes to.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; the mid-point quadrilateral proof is a typical 4 mark theorem.
Name the triangle in which you apply the mid-point theorem; do not just say "by the theorem".
Always say which diagonal is the common parallel.
For the median to the hypotenuse, mention that the right angle is at the vertex opposite.

Practice questions on Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse

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
$A B C$ is right-angled at $B$ and $A C = 13$ cm. The median $B M$ to $A C$ has length:
(a) 5 cm
(b) 6.5 cm
(c) 12 cm
(d) 13 cm
Hint
Half the hypotenuse.
Answer
(b) 6.5 cm
Q2 · 3 marks · Short answerHarder board level
Triangle $A B C$ has a right angle at $A$ , $A B = 5$ cm and $A C = 12$ cm. $D$ is the mid-point of $B C$ . Find $A D$ and $ar (△ A B D)$ .
Hint
$B C = 13$ ; median halves the area.
Answer
$B C = 13$ , $A D = 6.5$ cm; $ar (△ A B C) = 30$ cm², so $ar (△ A B D) = 15$ cm².
Q3 · 4 marks · Short answerHarder board level
$A B C D$ is a rectangle with $A B = 12$ cm and $B C = 8$ cm. $P, Q, R, S$ are the mid-points of its sides. Name the shape $P QR S$ , and find its side and area.
Hint
The diagonals are 12 and 8 across the mid-points.
Answer
$P QR S$ is a rhombus (the diagonals of a rectangle are equal). Its diagonals are $12$ and $8$ cm, so side $= 6^{2} + 4^{2} = 52 = 213$ cm and area $= \frac{1}{2} \times 12 \times 8 = 48$ cm², half the rectangle.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The mid-points of the sides of a rectangle form a rhombus.
Reason (R): The sides of the mid-point quadrilateral are half the diagonals, and the diagonals of a rectangle are equal.
(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
PQ = QR = half a diagonal.
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A ladder 13 m long leans against a vertical wall with its foot 5 m from the wall. A lantern hangs from its mid-point.
(a) How high up the wall does the ladder reach?
(b) How far is the lantern from the corner where wall and floor meet?
(c) The foot is slid out to 12 m. How high does the ladder reach and how far is the lantern from the corner now?
Hint
Median to the hypotenuse.
Answer
(a) 12 m (b) 6.5 m (c) 5 m high; the lantern is still 6.5 m from the corner.

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 equal intercept theorem.Show answer
Equal intercepts on one transversal by parallel lines mean equal intercepts on every transversal.
2.Length of the trapezium mid-line?Show answer
Half the sum of the parallel sides.
3.Shape formed by joining mid-points of any quadrilateral?Show answer
A parallelogram.
4.Median to the hypotenuse equals?Show answer
Half the hypotenuse.

Equal Intercepts, Trapezium Mid-points, Mid-point Quadrilateral and the Median to the Hypotenuse: frequently asked questions

Is the median to the hypotenuse in the ICSE Class IX syllabus?

Yes, it is listed in the mid-point and intercept theorems chapter.

Why is PQRS a parallelogram for any quadrilateral?

Because each pair of opposite sides is parallel to a diagonal and half of it.

When is PQRS a square?

When the diagonals of ABCD are equal and perpendicular.