Math Blog on Grade XI

Triangles:
Out of many geometrical structures, triangle is one of the basic shapes of geometry. Triangles are polygons having 3 sides or edges, which are nothing but line segments and 3 angles or vertices, which may be of same measures sometimes (for quick solution also try area of equilateral triangle calculator). The sides are termed as A, B, C and triangle is represented as Δ ABC.
As all grade XI board of intermediate education Andhra Pradesh students are aware, triangles are classified according to the lengths of the three sides:
1.      Isosceles triangle: Here, two sides are same in length and two angles which are opposite to the sides of same length are same in measure.

2.      Equilateral triangle: All sides are equal in length and all three angles are same measuring 60 ⁰ each (as total interior angle in a triangle is 180⁰) in case of equilateral triangles.


3.      Scalene triangle:All sides are different in length and all three angles are different in measure in case of scalene triangles.

Triangles are also classified according to the angles of three sides (for more on triangles visit this):
1.      Right angle triangle:These triangles has one of its interior angle measuring 90⁰ . In case of right triangles, special names are given to each side of the triangle, with the side which is opposite to the right angle being termed as  hypotenuse, which is the longest side of the right triangle and the other two sides being known as the legs.

2.      Acute  angle triangle: All three interior angles are less than 90⁰ in case of acute angle triangles.

 


3.      Obtuse angle triangle:These triangles have one angle which is more than 90⁰ .

In grade XI you will also be studying one more topic, i.e., special right triangles, which are nothing but right triangles having some specified features which makes calculation easy. There are two types in which special right triangle are being classified: angle based and side based.
In angle based special right triangle, the angles are divided such that the right angle (largest) is equal to sum of the other two angles. There are two types of  special right triangle which are angle based:
1.      45-45-90 angle:  As you can judge by name, it has 2 angles measuring 45⁰ and one 90⁰ and you can get such triangle by cutting a square diagonally. Here the angles are in the ratio 1 : 1 : 2 and by using Pythagoras theorem(which says - in a right angle triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the two legs.  The theorem is represented by the equation: a² + b² = c², where c represents the hypotenuse, longest side) , the sides are in the ratio of
1: 1 : √2, which means that two sides are equal in length also.
 Hence such kind of triangles have characteristic of both the isosceles and the right angle triangles.


2.      30-60-90 angle: This kind of special right triangles are very common and has one 30⁰ ,one 60⁰ and one 90⁰ angle.  There is one theorem which applies to these triangles, stated as,

In a 30-60-90 triangle, the  length of hypotenuse is two times the length of leg opposite to the 30^o angle. And the length of the other leg is calculated as SQRT (3) times the leg opposite to the 30^o angle.




Now finally, in side-based special right triangles, the lengths of the sides form ratios of whole number, such as 3 : 4 : 5 ,11 : 12 : 13 etc.

In upcoming posts we will discuss about Probability and Statistics and Congruence. Visit our website for information on Calculating Standard Deviation

How to Tackle Right Triangle Problems

Hello friends, today I am going to discuss the topic which is the most important topic for algebra 2 mathematic students of grade XI of maharashtra board. And that topic is “Special Right Triangles, Right Angles”. And this article is from one of the easiest in analytic geometry problems and trigonometric mathematics.
First we discuss about Special Right Triangle:- A special triangle is a sight angle triangle which makes the calculation on triangle easier .Right angle Triangle are those whose one angle is 90^â—¦ and the sum of other two triangle is 180°-90°=90°. side base right  angle triangle  are having length in the ratio of two whole number such as 1:2:3 , 37°-53°-90°.
Angle Base triangle is composed by the relationship of the angles. Angles in right angles are such that larger angle. Is equal to the sum of two other angle examples: - 30°-60°-90°, 45°-45°-90°.Trignometive functions are calculated by the special triangle.(want to Learn more about triangles ,click here),
Right Triangle whose sides are in geometric progression- A kepler triangle is triangle in geometric progression. as this sides are in the ration of
                                         a, ar, ar²
Common ratio r =q where q is golden ratio
Ratio = 1:√L: L
Now friends we discuss on Isosceles Right Triangle: - isosceles right triangles are those in which two angles and two sides are equal.  In isosceles right triangle Ratio of their sides is always 1:1:√2 or x x:x√2.

 Consider an example: - is one of the equal sides of isosceles triangle is -2 other side are
 As we know that ratio of side are       x x:x√2
                  x=2
                         2:2:2√2
It can also be solved by Pythagoras Thermos. in Pythagoras thermos an angle are such sum of the square of two side are equal to the square of larger side.
                          x²+y²=z²
               In isosceles triangle two side are equal
                  2²+2²= z²
                 4+4= z²
                 8=  z²
                    z= √8
           Z= √4*2= 2√2
Then discuss on Different type of triangle is Fibonacci triangle: - Fibonacci series is basically found in Pascal’s triangle. Normally Fibonacci series is the length of hypotenuse of a right triangle with integral side. We can say that  IN Pythagoras triple Fibonacci series is the large number of quantity.
IN Fibonacci series every number below in the triangle is the sum of the two numbers dynamically above it to the left and the right with position outside the triangle counting as zero.  Fibonacci triangle length of the longer leg  is equal to the sum of three side of the preceding triangle in  Fibonacci series of triangle.
Four most important numbers in the Fibonacci series is used to create a right triangle. With the base hypotenuse being determined by the second and third number and the other side being the square root of the product of the first and fourth number. Fibonacci number are (0, 1, 1, 2, 3, 8, 13) etc.
We discuss on properties and Attribute of right Triangle: -
Opposite side of the right angle this will always be the longest side of the right triangle.
When tow side of right triangle are not the hypotenuse. Then they are making up the two side right angle itself.
If the two sides include the right angle in equal length then right triangle can also be isosceles.
In above articles we discuss about the special right triangle and right angle and if anyone want to know about Special right triangles then they can refer to internet and text books for understanding it more precisely. You can also refer Grade XI  blog for further reading on Conditional Statements.