What is Section Formula?
Let’s first understand some basics of coordination chemistry. It is also known as Cartesian or Coordinate Geometry in classical mathematics and is the study of geometrical figures based on a coordinate system. In contrast, synthetic geometry is a purely mathematical concept. Mathematical analysis is utilised in physics and engineering. Most current areas of geometry, including algebraic, differential, discrete, and computational geometry, are built on top of this geometry. The Cartesian coordinate system is typically used to manage equations for planes, straight lines, and squares in two or three dimensions. The Euclidean plane (two dimensions) and Euclidean space are studied geometrically (three dimensions). Analytic geometry, as taught in school textbooks, is concerned with numerically defining and expressing geometrical forms, as well as extracting numerical information from those numerical definitions and representations. The Cantor–Dedekind axiom establishes that real-number algebra may be used to derive results concerning the linear continuum of geometry.
Distance formula and sector formula are two important concept of coordination geometry.
Distance between two points on the same axis of coordinates
The distance between two points on the same plane is represented by the difference between them. For example, if the points are on the y-axis, the difference between their ordinates determines the distance between them. Similarly, if the points are on the x-axis, the distance between them is represented by the difference in their abscissa.
Distance OP = 4 — (-1) = 5 units
Distance QR = 5 — (-3) = 8 units
Determining distance between Two Points Using Pythagoras Theorem
The Pythagoras Theorem, which defines the relationship between the sides of a right-angled triangle, is an important issue in mathematics. The Pythagorean Theorem is another name for it. You must have some fundamental knowledge of right-angle triangles before comprehending the formula and proof of Pythagoras theorem. “The square of the hypotenuse side of a right-angle triangle is equal to the sum of squares of the other two sides of the right-angle triangle,” according to the Pythagorean theorem. The Base, Perpendicular, and Hypotenuse are the three sides of this triangle. Because it is opposite the angle of 90°, the longest side is called the hypotenuse.
Calculating distance between two points using Pythagoras Theorem
Any two points on the cartesian plane can be B (c, d) and A (a, b).
To meet at C, draw lines parallel to the axes via B and A.
Δ BCA is right-angled at C.
With the help of the Pythagoras Theorem,
we get BA2 = BC2 + AC2
now we get (a — c)2 + (b — d)2
So,BA = √[a — c)2 + (b — d)2]
By making use of distant formula,
The distance between any two points (c, d) and (a, b) is given by
d = √[a — c)2+(b — d)2]
here d is the distance between the points (c, d) and (a, b).
The section formula gives the coordinates of P as: A(a, b) and B(c, d) internally in the ratio m:n.
P(x, y)=(mc+na/m+n ; md+nb/m+n)
To determine the ratio where a given point P(x, y) divides the line segment joining A(a, b) and B(c, d),
- We need to assume that the ratio is k : 1
- Now, we need to substitute the ratio in the section formula for any of the coordinates to get the value of k.
X = kc + a/k + 1
When the following a, c and x are known, we can calculate k. The same can be calculated from the y- coordinate also.
Any line segment is divided in half by its midway in the ratio 1:1.
The midpoint(P) of the line segment connecting A(a, b) and B(c, d) is given by
To determine the points of trisection P and Q that divide the line segment connecting A (a, b) and B (c, d) into three equal pieces, use the following formula:
i) AP: PB = 1: 2
P= (c+2a/3; d+2b/3)
ii) AQ: QB = 2: 1
Q= (2c + a/3; 2d + b/3)
Centroid of a triangle
If the vertices of an ABC are A (a, b), B (c, d), and C (e, f), then the coordinates of its centroid(P) are
p (x, y) = (a + c + e3, b + d + f3)
Area from Coordinates
The area of a triangle given its vertices
If the vertices of a Δ ABC are A(a, b), B(c, d) and C(e, f) then its area is given by
A = 1/2[a(d − f) + c(f − b) + e(b − d)]
Here A is the area of the Δ ABC.
If three points “P”, “Q” and “R” are collinear where Q lies between P and R, then we get,
- PQ + QR = PR. PQ, QR, and PR can be calculated utilizing the distance formula.
- The ratio in which Q divides PR, calculated utilizing section formula for both the x and y coordinates separately will be equal.
- Area of a triangle shaped by three collinear points is zero.