TNPSC – GEOMETRY
Geometry comes from the Greek meaning ‘earth measurement’ and is the visual study of shapes, sizes and patterns, and how they fit together in space.
That is, objects that have length (one dimension), length and width (two dimensions) and length, width and depth or height (three dimensions).
Dimensions of geometric objects: Point – No Dimensions. Line – One Dimension. Plane – Two Dimensions. Solid – Three Dimensions.
(i) Points: A Special Case: No Dimensions
A point is a single location on a flat surface. It is often represented by a dot on the page, but actually has no real size or shape.
You cannot describe a point in terms of length, width or height, so it is therefore non-dimensional. However, almost everything in geometry starts with the point, whether it’s a line, or a complicated three-dimensional shape.
(ii) Lines: One Dimension
A line is the shortest distance between two points. It has length, but no width, which makes it one-dimensional.
Wherever two or more lines meet, or intersect, there is a point, and the two lines are said to share a point:
Line segments and rays:
There are two kinds of lines: those that have a defined start- and endpoint and those that go on for ever.
Lines that move between two points are called line segments. They start at a specific point, and go to another, the endpoint. They are drawn as a line between two points, as you would probably expect.
The second type of line is called a ray, and these go on forever. They are often drawn as a line starting from a point with an arrow on the other end:
Parallel and perpendicular lines:
There are two types of lines that are particularly interesting and/or useful in mathematics. Parallel lines never meet or intersect. They simply go on forever side by side, a bit like railway lines:
Perpendicular lines intersect at a right angle, 90°:
Planes and Two-dimensional Shapes:
A plane is a flat surface, also known as two-dimensional. It is technically unbounded, which means that it goes on for ever in any given direction and as such is impossible to draw on a page.
One of the key elements in geometry is how many dimensions you’re working in at any given time. If you are working in a single plane, then it’s either one (length) or two (length and width). With more than one plane, it must be three-dimensional, because height is also involved.
Two-dimensional shapes include polygons such as squares, rectangles and triangles, which have straight lines and a point at each corner.
Two dimensional polygons, square, rectangle and triangle.
There are two ways by which we can classify triangles. One way is by determining the measures of a triangle’s angles. Another way in which triangles are classified is by the lengths of their sides. We will utilize both types of triangle classifications to aid in proofs throughout this section.
=> Classifying Triangles by Angles:
A triangle whose three angles are acute is called an acute triangle. That is, if all three angles of a triangle are less than 90°, then it is an acute triangle.
Every angle in these triangles is acute.
An obtuse triangle is a triangle that has one obtuse angle.
The obtuse angles in the triangles above are at vertex H and K, respectively.
A triangle that has one angle that is a right angle is called a right triangle. In other words, if one angle of a triangle is 90°, then it is a right triangle.
If all three angles of a triangle are congruent, then the triangle is an equiangular triangle. Later on, we will learn why the only angle measure possible for equiangular triangles is 60°.
Classifying Triangles by Sides:
A triangle with three congruent sides is called an equilateral triangle.
The tick marks indicate congruence between all three sides.
If a triangle has at least two congruent sides, then the triangle is an isosceles triangle. Note that, by definition, equilateral triangles can also be classified as isosceles.
A triangle that has no congruent sides is called a scalene triangle.
No two sides of the triangle above are congruent.
(iii) Three Dimensions: Polyhedrons and Curved Shapes
A two-dimensional shape has length and width. A three-dimensional solid shape also has depth. Three-dimensional shapes, by their nature, have an inside and an outside, separated by a surface. All physical items, things you can touch, are three-dimensional.
Polyhedrons (or polyhedra) are straight-sided solid shapes. Polyhedrons are based on polygons, two dimensional plane shapes with straight lines.
Polyhedrons are defined as having:
Flat sides called faces.
Corners, called vertices.
Polyhedrons are also often defined by the number of edges, faces and vertices they have, as well as whether their faces are all the same shape and size. Like polygons, polyhedrons can be regular (based on regular polygons) or irregular (based on irregular polygons). Polyhedrons can also be concave or convex.
One of the most basic and familiar polyhedrons is the cube. A cube is a regular polyhedron, having six square faces, 12 edges, and eight vertices.
Regular Polyhedrons (Platonic Solids):
The five regular solids are a special class of polyhedrons, all of whose faces are identical with each face being a regular polygon.
The platonic solids are:
=> Tetrahedron with four equilateral triangle faces.
=> Cube with six square faces.
=> Octahedron with eight equilateral triangle faces.
=> Dodecahedron with twelve pentagon faces.
=> Icosahedron with twenty equilateral triangle faces.
Tetrahedron Cube Octahedron Dodecahedron Icosahedron
Formula for Area (A) and Circumference (C):
Triangle: A = bh = x base x height
Rectangle: A = lw = length x width
Trapezoid: A = (b1 + b2)h = x sum of bases x height
Parallelogram: A = bh = base x height
Circle: A = πr2 = π * square of radius
C = 2πr = 2 * π * radius
C = πd = π * diameter
Formulas for Volume (V) and Surface Area (SA):
Rectangular Prism: V = lwh = length x width x height
SA = 2lw + 2hw + 2lh
= 2(length x width) + 2(height x width) + 2(length x height)
General Prisms: V = Bh = area of base x height
SA = sum of the areas of the faces
Right Circular Cylinder: V = Bh = area of base x height
SA = 2B + Ch = (2 x area of base) + (circumference x height)
Square Pyramid: V = 1/3 Bh = 1/3 * area of base * height
SA = B + 1/2 pl
= Area of base + (1/2 * perimetre of base * slant height)
Right Circular Cone: V = 1/3 Bh = 1/3 * area of base * height
SA = B + 1/2 Cl
= Area of base + (1/2 * Circumference * slant height)
Sphere: V = 4/3 πr3 = 4/3 * π * cube of radius
SA = 4πr2 = 4 * π * square of radius
Equations of a Line:
Standard Form: Ax + By = C, where A and B are not both zero
Slope-Intercept Form: y = mx + b or y = b + mx, where m = slope and b = y-intercept
Point-Slope Formula: y-y1 = m (x-x1), where m = slope and x1, y1 = point on line
Coordinate Geometry Formulas:
Let (x1,y1) and (x2,y2) be two points in the plane.
Slope = (y2 – y1) / (x2 – x1), where x2 = x1.
Midpoint = [(x1+x2)/2], [(y1 + y2)/2]
Distance = √[(x2-x1)2 + (y2-y1)2 ]
Distance Travelled: d = rt; distance = rate x time
Simple Interest: I = prt
interest = principal x interest rate x time
Polygon Angle Formulas:
Sum of degree measures of the interior angles of a polygon: 180 (n – 2)
Degree measure of an interior angle of a regular polygon: [180 (n – 2)] / n
where n is the number of sides of the polygon