Geometry Problem and Solution

Problem: Can two vertically opposite angles be supplement to each other? Draw a diagram to illustrate your answer


Let AB and CD bisect each other at right angles as shown in figure.







( Vertically opposite angles can be
supplements to each other)

Arithmetic Progressions

Arithmetic Progressions:
Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) – 3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
Each of the numbers in the list is called a term. In (i), each term is 1 more than the term preceding it. In (ii), each term is 30 less than the term preceding it. In (iii), each term is obtained by adding 1 to the term preceding it. In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding (or subtracting) 0 to the term preceding it. In (v), each term is obtained by adding – 0.5 to (i.e., subtracting 0.5 from) the term preceding it. In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression ( AP ). So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Slope of a Line

Slope of a Line:
A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
The angle (say) θ made by the line l with positive direction of x-axis and measured anti clockwise
is called the inclination of the line. Obviously 0° ≤ θ ≤ 180° (below Fig ). We observe that lines parallel to x-axis, or coinciding with x-axis, have inclination of 0°. The inclination of a vertical line (parallel to coinciding with y-axis) is 90°.


Definition 1 If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus, m = tan θ, θ ≠ 90° It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.

Perimeter and Area of a Circle

Perimeter and Area of a Circle:
circumference of a circle bears a constant ratio with its diameter. This constant ratio
is denoted by the Greek letter π (read as ‘pi’). In other words

Circumference/Diameter=π
circumference = π × diameter = π × 2r (where r is the radius of the circle) = 2πr


The great Indian mathematician Aryabhatta (A.D. 476 – 550) gave an approximate
value of π. He stated that π =62832/20000 which is nearly equal to 3.1416.
However, for practical purposes, we generally take the value of π as 22/7 or 3.14, approximately

area of the circle =1/2× 2πr × r = πr2

Introduction to Matrices

Introduction to Matrices:

The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of
matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment etc. Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial management.

Volume of cube calculation

Volume of cube calculation:
Problem:

A child playing with building blocks, which are of the shape of cubes, has built a structure as shown in Fig. below the edge of each cube is 3 cm, find the volume of the structure built by the child.


Solution:

Volume of a Cube = edge × edge × edge = a3

Solution : Volume of each cube = edge × edge × edge= 3 × 3 × 3 cm3
= 27 cm3 Number of cubes in the structure = 15 Therefore,
volume of the structure = 27 × 15 cm3 = 405 cm3


(here cm3 is cm to the power three)

How to find Surface area of a block?

How to find Surface area of a block?

The decorative block shown in below fig is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block.
(Take (pie π =22/7 ).

Solution : The total surface area of the cube = 6 × (edge)2 = 6 × 5 × 5 cm2 = 150 cm2. Note that the part of the cube where the hemisphere is attached is not included in the surface area. So, the surface area of the block = TSA of cube – base area of hemisphere + CSA of hemisphere = 150 – πr2 + 2 πr2 = (150 + πr2) cm2

= 150 cm2+(22/7+4.2/2+4.2/2)cm2

= (150 + 13.86) cm2 = 163.86 cm2