Practical Geometry

Practical geometry is the art of applying geometric principles and techniques to solve real-world problems, design structures and create objects with precision and accuracy. From constructing buildings to creating artwork, practical geometry plays a crucial role in various fields. In this chapter, we will explore the significance and practical applications of geometry.

Quadrilaterals

A quadrilateral is a closed geometric figure formed by connecting four line segments. It has four sides, four angles and four corners or vertices. Various categories of quadrilaterals are exemplified as follows:

Angle Sum Property of a Quadrilateral

Angle sum property of quadrilaterals states that the sum of all the angles within a quadrilateral is always equal to 360°.

∠ A + ∠ B + ∠ C + ∠ D = 360°

Adjacent and Opposite Sides

In a quadrilateral, there are two types of relationships between sides:

a. Adjacent Sides: Adjacent sides are two sides that meet at a shared corner or vertex. They are essentially like neighbours. In the case of the quadrilateral ABCD, the adjacent side pairs are:

→ AB and BC
→ BC and CD
→ CD and DA
→ DA and AB

b. Opposite Sides: Opposite sides are pairs of sides that do not meet at a common corner or vertex. They are positioned at the opposite ends of the quadrilateral. In the quadrilateral ABCD provided, the opposite side pairs are:

→ AB and DC
→ AD and BC

Convex and Concave Quadrilaterals

A convex quadrilateral is a quadrilateral that has all of its angles smaller than 180°.

A concave quadrilateral is a quadrilateral that has at least one angle larger than 180°.

Types of Quadrilaterals and their Properties

Circle

A circle is the collection of all points lying in a two-dimensional plane that share an equal distance from a fixed central point. This central point is denoted as the circle's centre and the constant distance from the centre to any point along the circle's boundary is called the radius.

Circumference of a circle: The circumference of a circle is the measurement of the circle's outer boundary. It is also known as the perimeter of a circle. As it signifies a length, it is quantified in units of measurement such as feet, inches, centimetres, metres, kilometres, etc.

Where,

π = ²²⁄₇ or 3.14

r = radius of the circle

Chord: A chord within a circle is a straight line segment that links two points on the circle's edge. In the given figure, line segment AB is a chord.

Diameter: The diameter of a circle is a unique chord that goes through the circle's centre. This diameter is the longest chord and is exactly twice the length of the radius.

Diameter = 2 × Radius

Radius = Diameter/2

In the given figure, AB is the diameter and OA is the radius.

Secant: A secant of a circle is a straight line that intersects the circle and touches it at two different points. In the given figure, line AB is a secant.

Arc: An arc of a circle is a segment that comprises a portion of the circle's outer boundary. In the given figure, AC, CB, BD and DA are arcs.

Sector: A sector of a circle is the region enclosed by an arc and two radii that extend from the endpoints of the arc to the centre of the circle.

Segment:  A segment of the circle is a region separated by a chord of a circle that divides the circular region into two parts.

Semicircle: A semicircle is a two-dimensional geometric shape that is half of a full circle. It is defined by a diameter that connects two points on the circumference of a circle and divides the circle into two equal parts.

Concentric Circles: Concentric circles are circles that have a common centre point but differ in their radii or the distance from the centre to their respective circumferences. Examples of Concentric circles are as follows:

Complementary Angles: Complementary angles are a pair of angles whose measures add up to 90°. Some examples of complementary angles are as follows:

Supplementary Angles: Supplementary angles are a pair of angles whose measures add up to 180°. Some examples of supplementary angles are as follows:

Angles Formed by a Transversal: In the given figure when lines l and m are intersected by the transversal p, there are eight angles formed and labelled as 1 to 8, each with specific names as listed below in the table.

When a transversal intersects two parallel lines:

→ Each pair of corresponding angles is equal.
→ Each pair of alternate interior angles is equal.
→ Each pair of interior angles on the same side of the transversal form. supplementary pairs.
→ Each pair of alternate exterior angles is equal.
→ Each pair of exterior angles on the same side of the transversal form supplementary pairs.

Polygons

A polygon is a closed shape bounded by three or more line segments. These segments are referred to as the sides of the polygon.

The point where two adjacent sides meet is called a vertex.

A diagonal is a line segment that connects two non-adjacent vertices within the polygon.

In the given figure, AB, BC, CD, DE, and EA represent the sides of the polygon, while A, B, C, D, and E are the vertices. AC, AD, BD, BE, CA, and CE are the diagonals of the polygon.

Types of Polygons: There are two main types of polygons − Regular polygons and Irregular polygons.

Regular Polygon: A polygon in which all sides have equal lengths and all angles have equal measures is called a regular polygon.

Irregular Polygon: A polygon with sides of unequal lengths and angles of unequal measures is called an irregular polygon.

Exterior Angle: An exterior angle is any angle formed outside a polygon. It is created by extending one of the polygon's sides beyond the point of intersection.

Sum of exterior angles in a Polygon = 360°

Interior Angle: An interior angle is any angle formed inside a polygon.

Sum of Interior Angles of a Polygon: The sum of the interior angles of a polygon with n sides can be calculated using the formula-

Sum of Interior Angles in a Polygon = (n − 2) × 180°

Sum of an interior angle and an exterior angle: The sum of an interior angle and an exterior angle is 180°.

Important Formulae to Remember