Triangle Inequalities: Sides and Angles

Let us learn what are Triangle inequalities,
You have just seen that if a triangle has equal sides, the angles opposite these sides are equal, and if a triangle has equal angles, the sides opposite these angles are equal. There are two important theorems involving unequal sides and unequal angles in triangles. They are:

Theorem: If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side.

Theorem: If two angles of a triangle are unequal, then the measures of the sides opposite these angles are also unequal, and the longer side is opposite the greater angle.

Example : Figure 1 shows a triangle with angles of different measures. List the sides of this triangle in order from least to greatest.
Figure : List the sides of this triangle in increasing order.

Because 30° < 50° < 100°, then RS < QR < QS.

Coordinates of a Point in Space

Let us learn about Coordinates of a Point in Space,
Having chosen a fixed coordinate system in the
space, consisting of coordinate axes, coordinate
planes and the origin, we now explain, as to how,
given a point in the space, we associate with it three coordinates (x,y,z) and conversely, given a triplet of three numbers (x, y, z), how, we locate a point in the space.
Given a point P in space, we drop a
perpendicular PM on the XY-plane with M as the
foot of this perpendicular (Fig.). Then, from the point M, we draw a perpendicular
ML to the x-axis, meeting it at L. Let OL be x, LM be y and MP be z. Then x,y and z
are called the x, y and z coordinates, respectively, of the point P in the space. In
Fig, we may note that the point P (x, y, z) lies in the octant XOYZ and so all x, y,
z are positive. If P was in any other octant, the signs of x, y and z would change accordingly. Thus, to each point P in the space there corresponds an ordered triplet
(x, y, z) of real numbers.
Hope the above explanation helped you, now let me give you some examples on Coordinates of a Point in Space.

Coordinate Axes and Coordinate Planes in Three Dimensional Space

Let us study about Coordinate Axes and Coordinate Planes in Three Dimensional Space.
Consider three planes intersecting at a point O such that these three planes are mutually perpendicular to each other (Fig). These
three planes intersect along the lines X′OX, Y′OY
and Z′OZ, called the x, y and z-axes, respectively.
We may note that these lines are mutually
perpendicular to each other.

These lines constitute the rectangular coordinate system. The planes XOY, YOZ and ZOX, called, respectively the XY-plane, YZ-plane and the ZX-plane, are known as the three coordinate planes. We take the XOY plane as the plane of the paper and the line Z′OZ as perpendicular to the plane XOY. If the plane of the paper is considered
as horizontal, then the line Z′OZ will be vertical. The distances measured from
XY-plane upwards in the direction of OZ are taken as positive and those measured
downwards in the direction of OZ′ are taken as negative. Similarly, the distance
measured to the right of ZX-plane along OY are taken as positive, to the left of
ZX-plane and along OY′ as negative, in front of the YZ-plane along OX as positive
and to the back of it along OX′ as negative. The point O is called the origin of the
coordinate system. The three coordinate planes divide the space into eight parts known
as octants. These octants could be named as XOYZ, X′OYZ, X′OY′Z, XOY′Z,
XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′. and denoted by I, II, III, ..., VIII , respectively.
Hope the above explanation helped you, now let me explain you about Navigational Coordinates.

Set builder method

Let me explain you about Set builder method, one of the method of representing a set.
In set-builder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set. For example, in the set
{a, e, i, o, u}, all the elements possess a common property, namely, each of them
is a vowel in the English alphabet, and no other letter possess this property. Denoting
this set by V, we write
V = {x : x is a vowel in English alphabet}

It may be observed that we describe the element of the set by using a symbol x
(any other symbol like the letters y, z, etc. could be used) which is followed by a colon
“ : ”. After the sign of colon, we write the characteristic property possessed by the
elements of the set and then enclose the whole description within braces. The above
description of the set V is read as “the set of all x such that x is a vowel of the English
alphabet”. In this description the braces stand for “the set of all”, the colon stands for
“such that”. For example, the set
A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that x is a natural number and x lies between 3 and 10. Hence, the numbers 4, 5, 6, 7, 8 and 9 are the elements of the set A. If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set-builder form as follows: A= {x : x is a natural number which divides 42} B= {y : y is a vowel in the English alphabet} C= {z : z is an odd natural number} Hope the above explanation helped you, now let me give you some examples on Set builder method.

Similarity of Triangles

Let us learn what is meant by Similarity of Triangles,
You may say that triangle is also a polygon. So, we can state the same conditions for the similarity of two triangles. That is:
Two triangles are similiar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion).

Note that if corresponding angles of two triangles are equal, then they are known as equiangular triangles. A famous Greek mathematician Thales gave an important truth relating to two equiangular triangles which is as follows:
The ratio of any two corresponding sides in two equiangular triangles is always the same.
It is believed that he had used a result called the Basic Proportionality Theorem (now known as
the Thales Theorem) for the same.
Hope this explains about Similarity of Triangles.

Solution of a Quadratic Equation by Completing the Square

In this section, we shall study Solution of a Quadratic Equation by Completing the Square,

Consider the following situation:
The product of Sunita’s age (in years) two years ago and her age four years
from now is one more than twice her present age. What is her present age?
To answer this, let her present age (in years) be x. Then the product of her ages
two years ago and four years from now is (x – 2)(x + 4).

Therefore, (x – 2)(x + 4) = 2x + 1
i.e., x2 + 2x – 8 = 2x + 1
i.e., x2 – 9 = 0

So, Sunita’s present age satisfies the quadratic equation x2 – 9 = 0.
We can write this as x2 = 9. Taking square roots, we get x = 3 or x = – 3. Since
the age is a positive number, x = 3.
So, Sunita’s present age is 3 years.
Now consider the quadratic equation (x + 2)2 – 9 = 0. To solve it, we can write
it as (x + 2)2 = 9. Taking square roots, we get x + 2 = 3 or x + 2 = – 3.
Therefore, x = 1 or x = –5
So, the roots of the equation (x + 2)2 – 9 = 0 are 1 and – 5.
In both the examples above, the term containing x is completely inside a square,
and we found the roots easily by taking the square roots. But, what happens if we are
asked to solve the equation x2 + 4x – 5 = 0? We would probably apply factorisation to
do so, unless we realise (somehow!) that x2 + 4x – 5 = (x + 2)2 – 9.
So, solving x2 + 4x – 5 = 0 is equivalent to solving (x + 2)2 – 9 = 0, which we have
seen is very quick to do. In fact, we can convert any quadratic equation to the form
(x + a)2 – b2 = 0 and then we can easily find its roots. Let us see if this is possible.
Look at Fig. 4.2.
In this figure, we can see how x2 + 4x is being converted to (x + 2)2 – 4.

Mathematical Modelling

First let us see why mathematical modelling is important,
Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc. Sometimes we solve the problems without going into the
physical insight of the situational problems. Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values. To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling. Let us consider the
following problems:
(i) To find the width of a river (particularly, when it is difficult to cross the river).
(ii) To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc.).
(iii) To find the height of a tower (particularly, when it is not possible to reach the top
of the tower).
(iv) To find the temperature at the surface of the Sun.
Appendix 2
(v) Why heart patients are not allowed to use lift? (without knowing the physiology
of a human being).
(vi) To find the mass of the Earth.
(vii) Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it).
(viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely).
(ix) Estimate the population of India in the year 2020 (a person is not allowed to wait
till then).
All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling. In fact, you might have studied the methods
for solving some of them in the present textbook itself. However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling.