Quadrilaterals
Quadrilaterals are four-sided polygons, and CAT loves them because a single figure can hide a property test, an area computation and a ratio twist all at once. The whole family descends from one parent — the parallelogram — and every special case (rectangle, rhombus, square) is just a parallelogram with an extra constraint on its sides, angles or diagonals. Get that hierarchy straight and most "which statement must be true" questions answer themselves. The trapezium and kite sit slightly apart: they are not parallelograms, but their area formulas and diagonal behaviour show up often in mensuration and coordinate-geometry sets. This chapter walks through each shape’s defining properties, its area formula, and the diagonal and angle facts that separate look-alikes — for instance, why a rhombus and a square both have perpendicular diagonals but only the square has equal ones. Expect questions that mix the half-diagonal-product area rule with the Pythagoras theorem, or that ask you to deduce a side from a given area and a diagonal. We build the property map first, then the area toolkit, then the CAT-style applications that reward students who reason from properties rather than memorise blindly.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Climb the hierarchy: parallelogram → (add right angles) rectangle, → (add equal sides) rhombus, → (add both) square. Every child keeps the parent’s properties.
- Equal diagonals ⇒ rectangle (or square). Perpendicular diagonals ⇒ rhombus or kite. BOTH equal AND perpendicular ⇒ square only.
- Rhombus, kite and square all use Area = ½ d₁ d₂. For a square d₁ = d₂, so it shrinks to ½ d².
- Rhombus side = ½√(d₁² + d₂²): the half-diagonals form a right triangle — pair it with Pythagorean triples (3-4-5, 5-12-13, 8-15-17).
- Isosceles-trapezium area: drop perpendiculars, the overhang each side = (longer − shorter)/2, recover height by Pythagoras, then ½(a + b)h.
- Parallelogram diagonal-sum law d₁² + d₂² = 2(a² + b²) recovers a missing diagonal from the sides.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Using side × side for a parallelogram’s area instead of base × perpendicular height.
- Assuming a parallelogram’s diagonals are equal or perpendicular — they are neither unless it is a rectangle/rhombus.
- Forgetting that only ONE diagonal of a kite is bisected, while in a rhombus BOTH diagonals bisect each other.
- Mixing up the trapezium formula: it is ½(a + b)h, the AVERAGE of the parallel sides times height, not their sum.
- Treating any quadrilateral with perpendicular diagonals as a square — it could be a rhombus or a kite.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Quadrilaterals when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Area formulas
| Parallelogram | Area = base × height |
|---|---|
| Rectangle | Area = length × breadth |
| Rhombus (diagonals) | Area = ½ × d₁ × d₂ |
| Square | Area = side² = ½ × diagonal² |
| Trapezium | Area = ½ × (a + b) × h |
| Kite | Area = ½ × d₁ × d₂ |
Diagonal & angle properties
| Rectangle diagonal | d = √(l² + b²) |
|---|---|
| Square diagonal | d = side × √2 |
| Rhombus side | side = ½ × √(d₁² + d₂²) |
| Parallelogram angles | adjacent angles sum to 180° |
| Diagonal sum law (parallelogram) | d₁² + d₂² = 2(a² + b²) |