Understand Pythagorean triples

In Geometry Tutoring we know that a natural number, n , is said to be the perfect square of the natural number, a, if n = a*a . In other words, we can say that a natural number is called a perfect square if it is the square of some other natural number. By the term square of a number, we mean that the product of a number multiplied by itself.
Then we have also learnt some properties of perfect squares. Adding to such list of properties, we will learn one important property of perfect squares here. This property of perfect squares is called the Pythagorean triples. Let’s define the term first.
Three natural numbers m, n, p are said to to form a Pythagorean triples if m  Ì‚2 + n  Ì‚2 = p  Ì‚2
The Pythagorean triples formula comes from the Phythagorean Theorem of right triangles in which the square of the length of hypotenuse is equal to the sum of the squares of the other two sides. In Pythagoras theorem, given a, b to be the two sides of a right triangle & h as the hypotenuse of the triangle, we have the relation h  Ì‚2 = a Ì‚2 + b Ì‚2. According to Pythagorean triples formula , if m is a natural  number greater than 1 , i.e., m>1, we can find a Pythagorean triples using the formula ,
( 2m , m  Ì‚2 – 1 , m  Ì‚2 +1 )
Lets take an example from Andhra Pradesh Board sample papers

Putting m = 4 , we get the Pythagorean triples as
2m = 2 * 4 = 8
m  Ì‚2 – 1 = 16 – 1 = 15
m  Ì‚2 + 1 = 16 + 1 = 17
Thus , the numbers 8, 15, 17 form the Pythagorean triples. For more information read here

This was all about Pythagorean triples. Visit our blogs for more information on Perpendicular Linesplanes and pythagorean theorem in grade vii

trigonometric ratios

Hello students, in this blog we are going to discuss the trigonometric ratios. The trigonometric ratios of angles are used to find the angles and sides for a triangle when we have some of angles and sides are given. This task is the main task of the trigonometric ratios. This problem is solved by using some ratios of the sides of a triangle with respects to its acute angles. These ratios of acute angles (What is an Acute Angle) are called the trigonometric ratios of angles.
Note: - The trigonometric ratios are same for the same length.
We have six trigonometric ratios; let’s take a look on them.
With reference to an angle ‘A’ in a right angled, ΔABC, right angled at ‘c’.
‘a’ is the opposite side (Perpendicular)
‘b’ is the adjacent side (Base)
‘c’ is the hypotenuse.
The ratios of sides a / c, b / c, a / b, b / a, c / b, c / a have the following names, they are : -
a / c is called the sine of ‘A’, written as sin A.
b / c is called the co-sine of ‘A’, written as cos A.
a / b is called the tangent of ‘A’, written as tan A.
b / a is called the co-tangent of ‘A’, written as cot A.
c / b is called the secant of ‘A’, written as sec A.
c / a is called the co-secant of ‘A’, written as cosec A.
Thus, we have six trigonometrically ratios which must be memorized
sin A = perpendicular / hypotenuse,
cos A = base / hypotenuse,
tan A = perpendicular / base,
cosec A = hypotenuse / perpendicular,
sec A = hypotenuse / base,
cot A = base / perpendicular,


In upcoming posts we will discuss about reflections and Permutations and combinations. Visit our website for information on business studies class 12 CBSE syllabus