Class 9 Mathematic quadrilaterals All Formula & Concept Notes | Formula Sheet

Class 9 mathematics quadrilaterals typically cover the following formulas and concepts:

1. Types of Quadrilaterals

1. Parallelogram: A quadrilateral with opposite sides parallel and equal.
2. Rectangle: A parallelogram with all angles equal (90 degrees).
3. Rhombus: A parallelogram with all sides equal.
4. Square: A parallelogram that is both a rectangle and a rhombus.
5. Trapezium (or Trapezoid): A quadrilateral with at least one pair of parallel sides.
6. Kite: A quadrilateral with two distinct pairs of adjacent sides that are equal.

2. Properties of Quadrilaterals

1. Parallelogram:

   • Opposite sides are equal.
   • Opposite angles are equal.
   • Consecutive angles are supplementary (sum to 180°).
   • Diagonals bisect each other.

2. Rectangle:

   • All properties of a parallelogram.
   • All angles are right angles.
   • Diagonals are equal.

3. Rhombus:

   • All properties of a parallelogram.
   • All sides are equal.
   • Diagonals are perpendicular bisectors of each other.
   • Diagonals bisect the angles.

4. Square:

   • All properties of a rectangle and a rhombus.
   • All sides are equal, and all angles are right angles.
   • Diagonals are equal, perpendicular bisectors, and bisect the angles.

5. Trapezium:

   • One pair of opposite sides are parallel (the parallel sides are called bases).
   • The non-parallel sides are called legs.
   • The sum of the angles on the same side of the trapezium is supplementary (180°).

6. Kite:

   • Two distinct pairs of adjacent sides are equal.
   • One pair of opposite angles are equal.
   • Diagonals are perpendicular.
   • One of the diagonals bisects the other.

3. Formulas Related to Quadrilaterals

1. Sum of Interior Angles:

   • For any quadrilateral, the sum of the interior angles is $360^\circ$.

   Formula: $\text{Sum of interior angles} = (n-2) \times 180^\circ$ where $n$ is the number of sides (for a quadrilateral, $n=4$).

2. Area Formulas:

   • Parallelogram: $\text{Area} = \text{base} \times \text{height}$

   • Rectangle: $\text{Area} = \text{length} \times \text{width}$

   • Rhombus: $\text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$

   • Square: $\text{Area} = \text{side}^2$

   • Trapezium: $\text{Area} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$

   • Kite: $\text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$

4. Theorems Involving Quadrilaterals

1. Parallelogram Diagonal Theorem:

   • The diagonals of a parallelogram bisect each other.

2. Rhombus Diagonal Theorem:

   • The diagonals of a rhombus are perpendicular bisectors of each other.

3. Rectangle Diagonal Theorem:

   • The diagonals of a rectangle are equal in length.

4. Square Diagonal Theorem:

   • The diagonals of a square are equal in length, perpendicular, and bisect each other.

5. Trapezium Angle Theorem:

   • In a trapezium, the sum of the angles on the same side of the non-parallel sides is $180^\circ$.

6. Kite Diagonal Theorem:

   • One of the diagonals of a kite bisects the other and is perpendicular to it.

These concepts and formulas are foundational for understanding quadrilaterals and solving related problems.