SSC CAPFS and DP - quadrilaterals, regular polygons, properties of circle E-Book Study Material

Quadrilaterals, Regular Polygons, Properties of Circle

Quadrilaterals

A quadrilateral is a closed, two-dimensional shape with four straight sides and four vertices (corners). The sum of the interior angles of any quadrilateral is always 360 degrees. Quadrilaterals can be classified into several types based on their side lengths and angle properties. Understanding these properties is crucial for solving geometry problems in competitive exams.

Types of Quadrilaterals

Let's explore the common types of quadrilaterals:

Parallelogram: A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. Consecutive angles are supplementary (add up to 180 degrees). Diagonals bisect each other.
Rectangle: A parallelogram with four right angles (90 degrees). Opposite sides are equal and parallel. Diagonals are equal in length and bisect each other.
Square: A rectangle with all four sides equal in length. It is also a rhombus and a rectangle. All angles are 90 degrees. Diagonals are equal, bisect each other, are perpendicular, and bisect the angles.
Rhombus: A parallelogram with all four sides equal in length. Opposite angles are equal. Diagonals bisect each other at right angles and bisect the angles.
Trapezium (or Trapezoid): A quadrilateral with at least one pair of parallel sides. The other two sides are non-parallel. In an isosceles trapezium, the non-parallel sides are equal, and the base angles are equal.
Kite: A quadrilateral with two distinct pairs of equal-length adjacent sides. One pair of opposite angles are equal. The diagonals are perpendicular, and one diagonal bisects the other.

Properties and Formulas for Quadrilaterals

Key formulas and properties to remember:

Sum of Interior Angles: (n-2) * 180 degrees, where n is the number of sides. For a quadrilateral, n=4, so (4-2) * 180 = 360 degrees.
Area of a Parallelogram: base × height
Area of a Rectangle: length × width
Area of a Square: side² or (diagonal²)/2
Area of a Rhombus: (diagonal₁ × diagonal₂)/2
Area of a Trapezium: [(sum of parallel sides)/2] × height
Diagonals of a Square: d = s√2, where d is the diagonal and s is the side length.

Memory Trick for Quadrilaterals: Think of a Square as the "perfect" quadrilateral. All sides equal, all angles 90 degrees. Rectangles are stretched squares (angles 90, opposite sides equal). Rhombus are squeezed squares (sides equal, opposite angles equal). Parallelograms are rhombuses/rectangles without the strict angle/side rules. Trapeziums have just one parallel pair. Kites have adjacent equal sides.

Example Problem:

The angles of a quadrilateral are in the ratio 1:2:3:4. Find the measure of each angle.

Let the angles be x, 2x, 3x, and 4x. The sum of angles in a quadrilateral is 360 degrees. So, x + 2x + 3x + 4x = 360 10x = 360 x = 36 The angles are: 1x = 36 degrees 2x = 72 degrees 3x = 108 degrees 4x = 144 degrees

Regular Polygons

A polygon is a closed shape made up of straight line segments. A regular polygon is a polygon that is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal in measure).

Properties of Regular Polygons

The properties of regular polygons are consistent and depend on the number of sides (n).

Sum of Interior Angles: (n-2) * 180 degrees.
Measure of Each Interior Angle: [(n-2) * 180] / n degrees.
Sum of Exterior Angles: Always 360 degrees for any polygon.
Measure of Each Exterior Angle: 360 / n degrees.
Relationship between Interior and Exterior Angle: Each interior angle + each exterior angle = 180 degrees.

Types of Regular Polygons

Some common regular polygons include:

Equilateral Triangle: 3 sides (n=3). Each interior angle = (3-2)*180/3 = 60 degrees.
Square: 4 sides (n=4). Each interior angle = (4-2)*180/4 = 90 degrees.
Regular Pentagon: 5 sides (n=5). Each interior angle = (5-2)*180/5 = 108 degrees.
Regular Hexagon: 6 sides (n=6). Each interior angle = (6-2)*180/6 = 120 degrees.
Regular Octagon: 8 sides (n=8). Each interior angle = (8-2)*180/8 = 135 degrees.

Formula Shortcut for Interior Angles: If you know the exterior angle (360/n), the interior angle is simply 180 - exterior angle. This is often quicker than using the (n-2)*180 formula.

Area of a Regular Polygon

The area of a regular polygon can be calculated using the formula: Area = (1/2) × perimeter × apothem Where: - Perimeter = n × side length (s) - Apothem (a) is the perpendicular distance from the center of the polygon to the midpoint of a side.

For specific polygons, there are dedicated area formulas:

Equilateral Triangle: (√3/4) × s²
Square: s²
Regular Hexagon: (3√3/2) × s²

Example Problem:

Find the measure of each interior angle of a regular octagon.

For a regular octagon, n = 8. Using the formula for each interior angle: [(n-2) * 180] / n = [(8-2) * 180] / 8 = (6 * 180) / 8 = 1080 / 8 = 135 degrees

Alternatively, using the exterior angle: Each exterior angle = 360 / n = 360 / 8 = 45 degrees. Each interior angle = 180 - exterior angle = 180 - 45 = 135 degrees.

Properties of Circle

A circle is a set of points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r). The diameter (d) is twice the radius (d = 2r) and is the longest chord passing through the center.

Key Terms Related to a Circle

Center: 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 with endpoints on the circle. d = 2r.
Chord: A line segment connecting any two points on the circle. The diameter is the longest chord.
Arc: A portion of the circumference of a circle.
Semicircle: An arc that is exactly half of the circle.
Circumference (C): The total distance around the circle. C = 2πr or C = πd.
Sector: The region bounded by two radii and the intercepted arc.
Segment: The region bounded by a chord and the intercepted arc.
Tangent: A line that touches the circle at exactly one point (the point of tangency).
Secant: A line that intersects the circle at two points.

Formulas for Circle

These formulas are fundamental for solving problems involving circles.

Area (A): A = πr²
Circumference (C): C = 2πr
Length of an Arc: (θ/360) × 2πr, where θ is the central angle in degrees.
Area of a Sector: (θ/360) × πr², where θ is the central angle in degrees.
Area of a Segment: Area of the corresponding sector - Area of the triangle formed by the radii and the chord.

Note: π (pi) is a mathematical constant approximately equal to 3.14159 or 22/7.

Key Circle Property: A radius drawn to the point of tangency is perpendicular to the tangent line. This is a very important property for solving problems involving tangents.

Properties of Chords

A diameter perpendicular to a chord bisects the chord and its corresponding arc.
Equal chords are equidistant from the center.
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.
Angles in the same segment of a circle are equal.
The angle in a semicircle is a right angle (90 degrees).

Example Problem:

A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the chord from the center.

Let the radius be OA = 10 cm. Let the chord be BC = 16 cm. Let M be the midpoint of BC. Then BM = MC = 16/2 = 8 cm. OM is the distance of the chord from the center. Triangle OMB is a right-angled triangle (since OM is perpendicular to BC). Using the Pythagorean theorem: OB² = OM² + BM² 10² = OM² + 8² 100 = OM² + 64 OM² = 100 - 64 OM² = 36 OM = √36 = 6 cm. The distance of the chord from the center is 6 cm.

Example Problem 2:

Find the area of a sector of a circle with radius 7 cm and a central angle of 60 degrees.

Radius (r) = 7 cm Central angle (θ) = 60 degrees Area of sector = (θ/360) × πr² = (60/360) × (22/7) × 7² = (1/6) × (22/7) × 49 = (1/6) × 22 × 7 = 154 / 6 = 77 / 3 cm² Approximately 25.67 cm².