SSC CGL - Circles and Quadrilaterals E-Book Study Material

Circles and Quadrilaterals

This section will cover two fundamental geometric shapes: Circles and Quadrilaterals. Understanding their properties, formulas, and theorems is crucial for solving various problems in geometry, which often appear in competitive exams. We will break down each topic into its core components, making it easier to grasp and remember.

Circles

A circle is a fundamental shape in geometry. It is defined as the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the center, and the fixed distance is called the radius.

Key Terminology Related to Circles

Before diving into properties, let's define some essential terms:

Center (O): The fixed point from which all points on the circle are equidistant.
Radius (r): The distance from the center to any point on the circle.
Diameter (d): A line segment passing through the center and connecting two points on the circle. It is twice the radius (d = 2r).
Circumference (C): The total distance around the circle. The formula is C = 2πr or C = πd.
Area (A): The space enclosed by the circle. The formula is A = πr².
Chord: A line segment connecting any two points on the circle. The diameter is the longest chord.
Secant: A line that intersects the circle at two distinct points.
Tangent: A line that touches the circle at exactly one point (the point of tangency).
Arc: A portion of the circumference of a circle.
Sector: The region bounded by two radii and the intercepted arc.
Segment: The region bounded by a chord and the intercepted arc.

Properties of Circles

Understanding these properties is key to solving circle-related problems:

All radii of the same circle are equal.
A diameter divides the circle into two semicircles.
The angle subtended by a diameter at any point on the circumference is a right angle (90°).
Equal chords subtend equal angles at the center.
The perpendicular from the center of a circle to a chord bisects the chord.
The line segment joining the center to the midpoint of a chord is perpendicular to the chord.
Chords equidistant from the center are equal, and vice versa.
A tangent to a circle is perpendicular to the radius through the point of contact.
The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal.
Opposite angles of a cyclic quadrilateral are supplementary (add up to 180°).

Formulas Related to Circles

Memorizing these formulas will be very helpful:

Circumference (C) = 2πr

Area (A) = πr²

Length of an arc of angle θ (in degrees) = (θ/360°) × 2πr

Area of a sector of angle θ (in degrees) = (θ/360°) × πr²

Area of a segment = Area of the corresponding sector - Area of the triangle formed by the radii and the chord.

Distance between two parallel tangents = Diameter (2r).

If two tangents are drawn from an external point to a circle, they are equally inclined to the line joining the point to the center, and they are of equal length.

Memory Trick: For circle formulas, remember π (pi) is your best friend. It's roughly 22/7 or 3.14. The circumference is a 'length' (2πr), and the area is a 'space' (πr²). Think of 'r squared' for area as it involves two dimensions.

Example Problem (Circle):

A circular garden has a radius of 21 meters. Find its circumference and area.

Given: Radius (r) = 21 meters.

Circumference (C) = 2πr = 2 × (22/7) × 21 = 2 × 22 × 3 = 132 meters.

Area (A) = πr² = (22/7) × (21)² = (22/7) × 21 × 21 = 22 × 3 × 21 = 1386 square meters.

Quadrilaterals

A quadrilateral is a polygon with four sides and four vertices. The sum of the interior angles of any quadrilateral is always 360°. Quadrilaterals can be classified into several types based on their side lengths and angle properties.

Types of Quadrilaterals

Let's explore the main types:

1. Parallelogram

A quadrilateral with opposite sides parallel.

Opposite sides are equal in length.
Opposite angles are equal.
Adjacent angles are supplementary (add up to 180°).
Diagonals bisect each other.

Area of a parallelogram = base × height.

Area of a parallelogram = ab sin(θ), where a and b are adjacent sides and θ is the angle between them.

2. Rectangle

A parallelogram with all four angles equal to 90°.

All properties of a parallelogram apply.
All angles are right angles (90°).
Diagonals are equal in length and bisect each other.

Area of a rectangle = length × width.

Diagonal (d) = √(length² + width²).

3. Square

A rectangle with all sides equal. It is also a rhombus with right angles.

All sides are equal.
All angles are right angles (90°).
Opposite sides are parallel.
Diagonals are equal, bisect each other, are perpendicular, and bisect the angles.

Area of a square = side².

Area of a square = (1/2) × (diagonal)².

Side = diagonal / √2.

4. Rhombus

A parallelogram with all four sides equal.

All sides are equal.
Opposite sides are parallel.
Opposite angles are equal.
Diagonals bisect each other at right angles (90°).
Diagonals bisect the angles of the rhombus.

Area of a rhombus = (1/2) × (product of diagonals).

Area of a rhombus = side² sin(θ), where θ is any interior angle.

5. Trapezium (or Trapezoid)

A quadrilateral with exactly one pair of opposite sides parallel.

The parallel sides are called bases.
The non-parallel sides are called legs.
The sum of adjacent angles between the parallel sides is 180°.

Area of a trapezium = (1/2) × (sum of parallel sides) × height.

6. Kite

A quadrilateral with two distinct pairs of equal-length adjacent sides.

One pair of opposite angles are equal.
The diagonals are perpendicular to each other.
One of the diagonals bisects the other.
One of the diagonals bisects the angles at its endpoints.

Area of a kite = (1/2) × (product of diagonals).

Memory Trick for Quadrilateral Areas:

Rectangle/Square: Simple length × width.
Parallelogram: Base × Height (think of slicing off a triangle and reattaching it to form a rectangle).
Trapezium: Average of parallel sides × Height.
Rhombus/Kite: Half the product of diagonals (think of splitting them into triangles).

Cyclic Quadrilaterals

A quadrilateral is called cyclic if all four of its vertices lie on a circle.

The key property is that the sum of opposite angles is 180°. (∠A + ∠C = 180° and ∠B + ∠D = 180°).
This property is derived from the theorem: "The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle."

Example Problem (Quadrilateral):

The diagonals of a rhombus are 12 cm and 16 cm. Find its area and the length of its side.

Given: Diagonal d1 = 12 cm, Diagonal d2 = 16 cm.

Area of rhombus = (1/2) × d1 × d2 = (1/2) × 12 cm × 16 cm = 6 cm × 16 cm = 96 cm².

In a rhombus, diagonals bisect each other at right angles. So, we have four right-angled triangles. The legs of each triangle are half the diagonals: 12/2 = 6 cm and 16/2 = 8 cm. The hypotenuse of this triangle is the side of the rhombus.

Using the Pythagorean theorem (a² + b² = c²): Side² = (6 cm)² + (8 cm)² = 36 cm² + 64 cm² = 100 cm².

Side = √100 cm² = 10 cm.

Relationship between Circles and Quadrilaterals

As mentioned, a quadrilateral whose vertices lie on a circle is a cyclic quadrilateral. The properties of cyclic quadrilaterals are a direct link between these two geometric shapes.

For example, if you have a square inscribed in a circle, the diagonals of the square are diameters of the circle. If you have a rectangle inscribed in a circle, its diagonals are also diameters.

Conversely, a circle can be inscribed in a quadrilateral (making it a tangential quadrilateral) if and only if the sums of opposite sides are equal (a + c = b + d). Examples include squares and rhombuses.

Key Theorems to Remember

Tangent-Radius Theorem: A tangent at any point of a circle is perpendicular to the radius through the point of contact.
Angle in a Semicircle Theorem: The angle subtended by a diameter at any point on the circumference is a right angle.
Angles in the Same Segment Theorem: Angles subtended by the same arc at the circumference are equal.
Cyclic Quadrilateral Theorem: Opposite angles of a cyclic quadrilateral are supplementary.
Chord Properties: Perpendicular from the center bisects the chord; equal chords are equidistant from the center.

Practice Tips

When solving problems involving circles and quadrilaterals:

Always draw a diagram.
Label all known information and angles/sides.
Identify the type of quadrilateral and its properties.
Look for right-angled triangles, as the Pythagorean theorem is frequently used.
Remember the relationship between angles at the center and circumference for circles.
Check if the quadrilateral is cyclic.
Use the correct formulas for area and perimeter.

Mastering the definitions, properties, and formulas for circles and quadrilaterals will provide a strong foundation for geometry problems in your exam.