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Class 9 · Chapter 10 · Geometry unit (25 of 80 marks)

4-gons (Quadrilaterals) Class 9: notes and important questions

Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 17 questions
  • 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
  • About 9 hours to master

Geometry unit: 25 of 80 theory marks (Introduction to Euclid's Geometry: Axioms and Postulates, Lines and Angles, Triangles – Congruence Theorems, 4-gons (Quadrilaterals), Circles).

In the NCERT book Ganita Manjari: Part II, Chapter 12, Quadrilaterals.

Revision notes

4-gons (Quadrilaterals): revision notes

1. Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, adjacent angles are supplementary, and the diagonals bisect each other. A diagonal divides it into two congruent triangles.

A quadrilateral is a parallelogram if any one of these holds: both pairs of opposite sides equal; both pairs of opposite angles equal; diagonals bisect each other; one pair of opposite sides both equal and parallel.

2. Special parallelograms

  • Rectangle: a parallelogram with a right angle; its diagonals are equal.
  • Rhombus: all sides equal; its diagonals bisect each other at right angles (so side\(^2 = \left(\dfrac{d_1}2\right)^2 + \left(\dfrac{d_2}2\right)^2\)).
  • Square: a rectangle and a rhombus; diagonals equal and perpendicular.

3. Mid-point theorem and its converse

Mid-point theorem: the segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. Converse: a line through the mid-point of one side, parallel to a second side, bisects the third side.

So the triangle formed by joining the mid-points of the three sides has half the perimeter and a quarter of the area of the original.

4. Centroid

A median joins a vertex to the mid-point of the opposite side. The three medians meet at one point, the centroid \(G\), which divides each median in the ratio \(2 : 1\) from the vertex: \(AG = \dfrac23AD\), \(GD = \dfrac13AD\). A thin triangular plate balances at \(G\). In coordinates, \(G\) is the average of the three vertices.

5. The mid-point quadrilateral

Joining the mid-points of the sides of any quadrilateral \(ABCD\), in order, gives a parallelogram: each side is parallel to a diagonal of \(ABCD\) and half as long. Its perimeter equals \(AC + BD\) and its area is half the area of \(ABCD\). If the diagonals of \(ABCD\) are equal it is a rhombus; if they are perpendicular, a rectangle; if both, a square.

Worked example 1

In parallelogram \(PQRS\), \(\angle P = (3x + 10)^\circ\) and \(\angle Q = (2x + 20)^\circ\). Find \(x\).

Adjacent angles: \(5x + 30 = 180 \Rightarrow x = 30\); \(\angle P = 100^\circ\), \(\angle Q = 80^\circ\).

Worked example 2

\(D\) and \(E\) are the mid-points of \(AB\) and \(AC\), and \(BC = 18\) cm. Find \(DE\).

\(DE = \dfrac12BC = 9\) cm, and \(DE \parallel BC\).

Worked example 3

A rhombus has diagonals \(10\) cm and \(24\) cm. Find its side.

Half-diagonals \(5\) and \(12\) meet at right angles: side \(= 13\) cm.

Common errors

  • Taking adjacent angles of a parallelogram as equal: they are supplementary.
  • Assuming the diagonals of every parallelogram are equal or perpendicular.
  • Using the ratio \(1 : 2\) for the centroid from the vertex: it is \(2 : 1\).
  • Using the mid-point theorem when only one point is a mid-point: use the converse, with the parallel line.

Exam tips

  • Name the property or condition you use: "(opposite sides of a parallelogram)", "(mid-point theorem)".
  • To prove a quadrilateral is a parallelogram, one condition from section 1 is enough: say which one.

Topics in this chapter: Parallelograms · Mid-point theorem · Special parallelograms · Centroid · The mid-point quadrilateral.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can use the angle and side properties of parallelograms, the mid-point theorem and the centroid ratio.

Read first: 1. Parallelograms; 2. Special parallelograms; 3. Mid-point theorem; 4. Centroid 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%
Practise step 1
Step 2

Exam standard

You can find unknown sides and angles and prove short results about parallelograms and mid-points.

Read first: 1. Conditions for a parallelogram; 3. Mid-point theorem; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can prove the mid-point theorem and work with the mid-point quadrilateral in coordinates and in words.

Read first: 3. Mid-point theorem (proof); 5. The mid-point quadrilateral 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can use the converse of the mid-point theorem in trapeziums and classify mid-point quadrilaterals.

Read first: 3. Converse of the mid-point theorem; 5. Special mid-point quadrilaterals 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceParallelograms

In parallelogram \(ABCD\), \(\angle A = 70^\circ\). Then \(\angle B\) is

  1. (a)\(70^\circ\)
  2. (b)\(140^\circ\)
  3. (c)\(20^\circ\)
  4. (d)\(110^\circ\)
Q2·1 mark·Multiple choiceMid-point theorem

\(D\) and \(E\) are the mid-points of sides \(AB\) and \(AC\) of \(\triangle ABC\), and \(BC = 14\) cm. Then \(DE\) is

  1. (a)\(28\) cm
  2. (b)\(14\) cm
  3. (c)\(7\) cm
  4. (d)\(3.5\) cm
Q3·1 mark·Multiple choiceSpecial parallelograms

The diagonals of a rhombus are \(16\) cm and \(12\) cm. Its side is

  1. (a)\(10\) cm
  2. (b)\(14\) cm
  3. (c)\(20\) cm
  4. (d)\(8\) cm

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I113 · Problem I120 · Problem I23 · Problem I105. All 29 →

Where marks are lost in 4-gons (Quadrilaterals)

  • Adjacent angles of a parallelogram set equal. Fix: adjacent angles add to 180°; opposite angles are equal.
  • Properties of special parallelograms used for every parallelogram (equal or perpendicular diagonals). Fix: check which figure it is.
  • Centroid ratio reversed. Fix: AG : GD = 2 : 1, the longer part next to the vertex.
  • Mid-point theorem quoted without both mid-points. Fix: with one mid-point and a parallel line, quote the converse.
  • Proof that a figure is a parallelogram without naming the condition. Fix: 'diagonals bisect each other, so ABCD is a parallelogram'.

Test 4-gons (Quadrilaterals) against the clock

Want it against the clock? Take a timed 30-mark chapter test on 4-gons (Quadrilaterals), new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Essentials or the free trial).