area of triangle, condition of co-linearity of three points. Trigonometry – Measure of angles in the sexagesimal. contesimal and circular systems

Area of triangle

The area of triangles can be found by using the following formula:

 

Area =  Base × height / 2

The red line represents the length of the base (call it b if you want) and the blue line represents the length of the height.

To get the area, just multiply the value of the base by the value of the height and divide the result by 2.

Example:

Find the area for the following figure:

The area is 18 square inches or 18 inches2.

 

Condition of collinearity of three points

Collinear points: Three points A, B and C are said to be collinear if they lie on the same straight line.

There points A, B and C will be collinear if AB + BC = AC as is clear from the adjoining figure.

In general, three points A, B and C are collinear if the sum of the lengths of any two line segments among AB, BC and CA is equal to the length of the remaining line segment, that is,

either AB + BC = AC or AC +CB = AB or BA + AC = BC.

In other words,

There points A, B and C are collinear if:

  • AB + BC = AC i.e., Or,
  • (ii) AB + AC = BC i.e. ,
  • Or, AC + BC = AB i.e.,

 

Measure of angles

In the sexagesimal

In Sexagesimal System, an angle is measured in degrees, minutes and seconds.

A complete rotation describes 360°. In this system, a right angle is divided into 90 equal parts and each such part is called a Degree (1°); a degree is divided into 60 equal parts and each such part is called a Sexagesimal Minute (1’) and a minute is further sub-divided into 60 equal parts, each of which is called a Sexagesimal Second (1’’). In short,

1 right angle    =  90 degrees (or 90°)

1 degree (or 1°)   = 60 minutes ( or 60’)

1 minute ( or 1’ ) =  = 60 seconds ( or 60’’)

 

In Centesimal System

In Centesimal System, an angle is measured in grades, minutes and seconds. In this system, a right angle is divided into 100.

equal parts and each such part is called a Grade (1g); again, a grade is divided into 100 equal parts and each such part is called a Centesimal Minute (1‵) ; and a minute is further sub-divided into 100 equal parts, each of which is called a Centesimal Second (1‶). In short,

1 right angle =  100 grades (or, 100g)

1 grade ( or 1g) = = 100 minutes (or, 100‵)

and 1 minute (or 1‵)            = = 100 seconds ( or, 100‶).

 

 

Circular System

In this System, an angle is measured in radians. In higher mathematics angles are usually measured in circular system. In this system a radian is considered as the unit for the measurement of angles.

Radian: A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius.

In any circle, the angle subtended at its centre by an arc of the circle whose length is equal to the radius of the circle is called a radian. Let OX = r be the radius of a circle having center at O

Now, take an arc XY of the circle such that arc XY = r and join OY. By definition, ∠XOY = one radian. One radian is written as 1c, 2 radians as 2c and in general, k radians as kc.

Circular (radian) measure of an angle:

The circular measure of an angle is the number of radians it contains.  Thus the circular (radian) measure of a right angle is π/2.

 

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