Friends in this session we are going to solve some questions and, we will discuss our approach
and thinking as when we see question how we recognise the topic and how to begin with,
all those things will be discussed while solving.
So lets start as if we are appearing for some test.
The question is, Are these points (2, 0), (2, -4), (5,-4) collinear?
Three points are given in terms of coordinates that mean its a question of co-ordinate geometry.
But we will discuss about the topic later for now just assume that any three points
or just three dots are there on a paper namely A,B & C. and suppose AB=3 units BC=8 units.
Can you tell me which three points are in a straight line i.e. collinear?
For that lets join all three points.
When we join these points it should make a straight line.
At only one place we are getting a straight line because at this point distance between
A and C is equal to the addition of the distances AB and BC.
At rest of the places it would make a triangle.
Since AB=3 units and BC=5 units then AC must be 8 units in order to be in a straight line
or else AC<8 units and forms a triangle.
Now we will find the distances between the coordinate points and check whether it forms
a straight line or not?
So here we proceed.
The given part is three points so lets name them, now these points are P, Q and R and
we have to see whether points are collinear are not?
And this is the tool which will help us to find the distances between the coordinate
points, friends you can write this in such manners as well order of X1, X2 and Y1, Y2
doesnt matter because when we square a negative or a positive number it gives us a positive
number only.
And this is the solution part.
The distance P and Q is 4 units, distance between QR and distance between PR.
I found the distances and noticed that addition of these two small numbers is not equal to
the remaining bigger one.
So I conclude that they are not collinear.
Friends you can see that before starting the solution we discussed a lot and after that
we solved it quickly without any mistake.
But as we discussed if points are not collinear they will form a triangle so friends can you
tell me what kind of triangle it is??
We recognise the type of triangles by looking at their sides whether all of the sides are
equal, only two sides are equal or all of them are different lengths.
So a) If all distances are equal it is an equilateral
triangle b) If only two distances are equal then it�s
an isosceles c) And the third case is the triangle may
be a scalene triangle as well.
In our case all the sides are of different lengths so it is neither equilateral nor isosceles
that means its a scalene triangle.
Friends a triangle may be a right angled triangle also so if we want to check whether our triangle
is a right angled triangle or not for that what must we do??
As we hear the name right angled triangle suddenly Pythagoras theorem pops up in our
mind.
So square all the values and see if sum of any two small numbers is equal to the remaining
bigger one or not.
So I found that its a scalene right angled triangle.
See how easy things are, we just need to concentrate a bit and apply a little bit of logic.
Alright lets quickly have a look at this question.
Are these points the vertices of a square??
Let me tell you the idea behind it.
First of all I will make an imaginary diagram of a quadrilateral ABCD.
Now if AB, BC, CD, and AD are equal means its a square right??
So we will find all the distances.
But still this is not the whole solution.
Even if all four distances are equal we cannot say that its a square because it may me a
rhombus as well.
Now what to do??
Another property of square is that its diagonals are also equal.
Therefore with all four sides we will also have to find whether diagonals AC and BD are
also equal or not.
Now this will be the proper solution.
You must be thinking if we directly find AC and BD the lengths of the diagonals only,
is it enough to check for a square?
The answer is no!
Because even rectangle and isosceles trapezium also have equal diagonals, so we need the
information of sides along with diagonals to confirm whether its square, rectangle or
trapezium.
So to find out whether the given pints are collinear, to recognise the different types
of triangles including the right angled triangle and to determine whether the four given coordinates
form a square we used only one formula i.e. the distance formula.
Before going take this question as an exercise and try to solve it and write your answer
in the comment box till we next time keep watching keep learning thank you.