Preliminaries in plane geometry (Part 1)

Many shapes in a plane are well represented by equations. Circles, parabolas and hyperbolas are some of the shapes which can be represented by equations.
But to deal with them we need to know the basics of geometry. The posts related to basics of coordinate geometry is given in the related posts "Preliminaries in plane geometry". The posts will be given in many parts. each part will cover two to three topics. In this post we will discuss about Distance formula and Section formula.

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Distance formula

The Cartesian coordinates are used to represent points in a plane. Every point in a place has a one to one relationship to a coordinate point. The distance (d)  between two points A(x₁, y₁) and B(x₂, y₂) is given by the formula
d = √[(x₂ − x₁)²+(y₂ − y₁)²]

The above formula can be found with the help of Pythagoras theorem. Draw a right angled triangle as shown in the graph. When we draw a right angled triangle we draw it such that the two edges including the right angle are parallel to the two axis. This helps us to find the coordinate of the vertex. Name the right angled vertex as C. The coordinate of the point C is (x₂, y₁). a line parallel to the coordinate axis has its other coordinate same. Suppose the line is parallel to x-axis then all the points on the line has its y-coordinate equal.  The distance between the points A and B is d. The distance between the points A and C is AC =(x₂ − x₁) and distance between the points B and C is BC =(y₂ − y₁). By Pythagoras Theorem we have (AB)² = (AC)² + (BC)². Substituting the value of AC and BC we get

d² = (x₂ − x₁)²+(y₂ − y₁)²Taking square root and as distance cannot be negative, we get
d = √[(x₂ − x₁)²+(y₂ − y₁)²]

Example:
Distance between A(3,5) and B(1,2) is
AB = d
d = √[(1 − 3)²+(2 − 5)²]
= √(4+9)
= √13

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Section Formula

A point C(x,y) is present between A(x₁,y₁) and B(x₂,y₂). C divides AB internally in the ratio m:n. Then the coordinate of C is given by

C≡( [mx₂+nx₁)/(m+n)] , [(my₂+ny₁)/(m+n)] )

Example:
C divides AB internally in the ratio 1:2. A(3,5) and B(1,2). Find C.
Let the coordinate of C be (x,y). Then
Here m=1, n=2, x₁= 3, x₂ = 1, y₁ = 5, y₂ = 2.
x = (1×1 + 2×3)/(2+1)
= 7/3
  y= (1×2 + 2×5)/(2+1)
= 12/3
= 4. Hence the coordinate of C is (7/3,4).

If C divides AB externally in the ratio m:n. Then the coordinate of C is given by

C≡( [mx₂− nx₁)/(m − n)] , [(my₂− ny₁)/(m − n)] )

Example:
C divides AB externally in the ratio 2:1. A(3,5) and B(1,2). Find C.
Let the coordinate of C be (x,y). Then
Here m=2, n=1, x₁= 3, x₂ = 1, y₁ = 5, y₂ = 2.
x = (2×1 − 1×3)/(2 − 1)
= −1
  y = (2×2 − 1×5)/(2 − 1)
= −1
Hence the coordinate of C is (−1,−1).

Right Angled Triangle

When we add the squares of 3 and 4 we get 5² or in other words 9+16=25. This property was first found as these numbers. Later with the help of geometry and algebra it was proved that the sides of a right angled triangle follow the rule a² + b² = c², where a,b and c are the length of the sides of the triangle. This theorem was later called as Pythagoras Theorem.

Let us derive this theorem.

The triangles BCD and ABC are similar
we have, BD/AB = DC/BC = BC/AC             (i)
Also triangles ABC and ADB are similar
we have, AD/AB = AB/AC = BD/CB             (ii)
As triangles ABC is similar to ADB.
From above AD/AB = AB/AC ;
AB² = AD·AC = (AC − DC)AC
=AC² − DC·AC       from (i)
=AC² − BC²

Pythagoras Theorem is in a triangle ABC
right angled at B
AB² + BC² = AC²

Triangles

Let us consider the persons who developed mathematics as one person. Let us call this person as Cofu. Then we could easily understand about the thoughts which was going in the minds of the people who developed mathematics. Human categorizes or differentiate things. They differentiated seasons in different ways and wrote them as tally marks on the walls of caves. Now considering all as cofu. Cofu looked at his surroundings and found many shapes among which was a shape with three sides. He called this shape as triangle. Then he wanted to rigorously study this shape. He found that the triangles may be classified according to the length of triangle's sides. If all the three sides are different then the triangle is called scalene triangle. If two sides are equal then the triangle is called isosceles. If all the sides are equal then the triangles is called equilateral.

As the time passed he developed the method to measure angles. He found that the angles are related to the sides or in other words they depend on each other. If we increase one angle then the side opposite to it also increases and if we increase the sides then the angles opposite to it also increases. If two angles are equal then the two sides are equal in a triangle.


He also differentiated triangles according to the angles. A triangle with one angle greater than 90° is called an obtuse angled triangle and those with all the angles less than a right angle is called an acute angled triangle. If any of the sides is equal to right angle it is called a right angle triangle.

As the time passed he wanted to measure angles. He faced a problem that there are many sizes of triangles and each shape has different relations of sides. After thinking a while he found that every triangle can be decomposed into two triangles with one angle of each triangle right angle. then he thought to develop all the mathematics of triangles according to right angle. We will study about the right angled triangles in the next post.