Excenters of a Triangle

An excenter of a triangle is a point at which the line bisecting one interior angle meets the bisectors of the two exterior angles on the opposite side. This is the center of a circle, called an excircle which is tangent to one side of the triangle and the extensions of the other two sides. Every triangle has three excenters and three excircles. The radius of excircle is called the exradius. 

In the following applet , the internal bisector of angle B of triangle ABC and bisectors of exterior angles at A and C meet at E₁. E₁ is one of the excenters of triangle ABC. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Similar Triangles and Medians

If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then the two triangles are similar.

Two triangles ABC and DEF , in which AP and DM are the medians , such that                AB/DE = AC/DF = AP/DM  , then ΔABC  ∼ ΔDEF 

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Construction - Isosceles Triangle - II

In continuation to my last post, this is the second method of construction of an isosceles triangle. This construction uses the fact that all the radii of a circle are equal. 

Following are the steps of construction : 

1. Draw a circle from with centre O. 
2. Take two points A and B on the circumference of circle. 
3. Draw line segment OA. 
4. Draw line segment OB. 
5. Draw line segment AB.
Triangle OAB is the required isosceles triangle. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Construction - Isosceles Triangle - I

A triangle in which two sides are equal is called an isosceles triangle. In an isosceles triangle , angles opposite to the equal sides are equal. 

The following construction of an isosceles triangle is based on the property that the perpendicular drawn, from the vertex where two equal sides meet, to the third side bisect the third side.

Following are the steps of construction :

1. Mark points A and B. 
2. Draw line segment AB. 
3. Draw the perpendicular bisector of segment AB. 
4. Mark any point C on the perpendicular bisector. 
5. Draw line segment AC. 
6. Draw line segment BC. 

Triangle ABC is the required isosceles triangle. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Exterior Angle Sum Property of a Triangle

A triangle has three corners, called vertices. The sides of a triangle that come together at a vertex form an angle. This angle is called the interior angle. In the figure below, the angles a,b and c are the three interior angles of the triangle. We know that the sum of interior angles of a triangle is 180°. You may visit the  links http://mathematicsbhilai.blogspot.in/2012/01/triangle-angle-sum-property-ii.html  and http://mathematicsbhilai.blogspot.in/2011/05/triangle-angle-sum.html

An exterior angle is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the figure, angle d is an exterior angle.

In the above figure ∠ACD is an exterior angle of Δ ABC. 

Because ∠a +∠b +∠c = 180°, and ∠b +∠d = 180°, we can see that that ∠d =∠a +∠c. This is stated as a theorem.  An exterior angle of a triangle is equal to the sum of the two opposite (nonadjacent) interior angles. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Hyperbolic Functions - II

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Hyperbolic Functions - I

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" and the hyperbolic cosine "cosh" from which are derived the hyperbolic tangent "tanh" and so on, corresponding to the derived trigonometric functions. 

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.

The hyperbolic functions cosh x and sinh x are defined using the exponential function e^x. Following are the definitions of cosh x , sinh x and tanh x.

cosh x = (e^x+ e^-x)/2

sinh x = (e^x - e^-x)/2

tanh x = (e^x - e^-x)/(e^x + e^-x)

Hyperbolic sine and cosine satisfy the identity cosh² - sinh²=1
Select the check box in the following applet to view graph of function.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Inverse Trigonometric Functions - II

In continuation of the previous post on Inverse Trigonometric Functions , let us define the remaining three functions.

•  The inverse cotangent function , denoted by cot^-1 is the function with domain R,range      (-π/2,0) U (0,π/2], defined by y=cot^-1x  => x = coty The inverse sine function is also called arccotangent , it is denoted by arccot. 
• The inverse secant function , denoted by sec^-1 is the function with domain       (-∞,-1] U [1, ∞), range [0,π/2) U (π/2,π] , defined by y = sec^-1x => x = secy The inverse secant function is also called arcsecant , it is denoted by arcsec. 
• The inverse cosecant function , denoted by cosec^-1 is the function with domain (-∞,-1] U [1, ∞), range [-π/2,0) U (0,π/2] , defined by y = cosec^-1x =>                x = cosecy The inverse cosecant function is also called arccosec , it is denoted by arccosec.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Criteria for Similarity of Triangles

Inverse Trigonometric Functions - I

For a function to have an inverse, it must be one-to-one. Sometimes, when a function is not one-to-one, one can restrict its domain to make it one-to-one. 

The inverse trigonometric functions are the inverse functions of the trigonometric functions. Because of their periodic nature, the trigonometric functions are not one-to-one. By restricting their domains, we can construct one-to-one functions from them. 

Remembering that the domain of a function and the range of its inverse are the same, we can define the following inverse trigonometric functions : 

• The inverse sine function , denoted by sin^-1 is the function with domain [-1,1] , range [-π/2,π/2] , defined by y = sin^-1x => x = siny .The inverse sine function is also called arcsine , it is denoted by arcsin. 
• The inverse cosine function , denoted by cos^-1 is the function with domain [-1,1] , range [0,π] , defined by y = cos^-1x => x = cosy The inverse cosine function is also called arccosine , it is denoted by arccos. 
• The inverse tangent function , denoted by tan^-1 is the function with domain R , range [-π/2,π/2] , defined by y = tan^-1x => x = tany The inverse tangent function is also called arctangent , it is denoted by arctan.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Signum Function

In mathematics, the signum function is an odd mathematical function that extracts the sign of a real number. The signum function is defined as

Signum function is also known as sign function.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Least Integer Function

The function whose value at any number x is the smallest integer greater than of equal to x is called the least integer function. It is denoted by ⌈x⌉ It is also known as ceiling of x. For example ⌈3.578⌉ = 4 , ⌈0.78⌉ = 1 , ⌈-4.64⌉ = - 4.

The graph of the least integer function lies on or above the line y = x , so it provides an integer ceiling for x.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Greatest Integer Function

The function, or rule which produces the "greatest integer less than or equal to the number" operated upon, is known as greatest integer function. It is denoted by symbol ⌊⌋ , i.e. ⌊x⌋≤x. It is also known as floor of x. Thus ⌊4.5778⌋=4 , ⌊0.75⌋=0 , ⌊-8.7275⌋=-9.

The Graph of the greatest integer function lies on or below the line y = x so it provides an integer floor for x.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Piecewise Functions

A piecewise function is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial functions: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

Note that, in this case, we have three different “formulas” for f(x): -x + 1, 2 and x²- 4. But remember, with a function, when you evaluate f(x), you should get ONE ANSWER—not three. Otherwise, you don’t have a function.

How do you know which answer you should have (i.e., which formula you should use)? The “formula” is chosen using the right-side of the piecewise-defined function (the “if …” part).

For example, suppose we wanted to find f( − 3). Since − 3 < − 1, we will use the first “formula”:

f( − 3) = - (-3) +1 = 4.

If we want to find f(5), since 5 > 3, we use the third “formula”:

f(5) = 5² - 4 = 21.

What about finding f(1)? Note that in the interval (-1,3) the value of function is 2.

The following applet shows the graph of piece wise function 

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Shifting Graphs

In this applet we will explore how to change an equation to shift its graph up or down or left or to the right or left. This can help us spotting familiar graphs in new positions. This will also help us graph unfamiliar equations more quickly.

To shift the graph of a function y=f(x) straight upwards, we add a constant to the right hand side e.g. by adding 2 to the right hand side of the formula y = x² we get y = x²+2 and the graph is shifted up by 2 units.

To shift the graph of equation y = f(x) straight down, we subtract we add a negative constant to the right hand side of the formula y = f(x)

To shift the graph of y = f(x) left or right , we add positive or negative constant to x respectively e.g. if by adding 2 to x in y = x² to get y = (x+2) ² , shifts the graph 2 units to the left.

In general we can say

•  y = f(x) + b , shifts the graph up by b units if b > 1 or shifts it down by |b| units if b <0 
•  y = f(x-a) , shifts the graph right a units if a > 0 or shifts it left |a| units if a<0. 

 Right Click any where on the graph and select ‘Trace On’ , then drag sliders ‘a’ or ‘b’.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Radical Functions

A radical function is any function that contains a variable inside a root. This includes square roots, cubed roots, or any nth root , for example f(x) = √(x+5) , g(x) = ∜(x-2) etc.

Here we will discuss about functions involving independent variable inside a square root sign or square root functions. In general the form of a square root function is                 f(x)=a√(x-b)+c , where a , b and c are some real numbers.

Graph of such a function is shown in the applet below.

To find domain of a square root function , the term inside the radical must be equal to or greater than zero, otherwise it is undefined. This means that only the x values that make the term inside the radical positive are defined and in the domain. 
For example , f(x) = √(x-4) + 3.

Since (x-4) is inside the radical, the domain lies on all the points where x makes (x-4) greater than or equal to zero. 
       x-4 ≥ 0
       x ≥ 4 

So the domain of the function is [4, ∞). 

The range of the function is then all the points of the y-axis that we get by putting x values of the domain. Let us start at the point x=4 and put it into the equation. f(x)=√(4-4)+3=3.

Now it can be easily concluded that by putting in any number greater than 4 for x we get f(x) larger than 3, so the smallest number in the range is 3. Thus we see that for any x value above 4 the function is defined. Therefore the range is [3,∞).


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Even - Odd Functions

Every function can be classified as an even function, an odd function, or neither. Even functions have the characteristic that f (x) = f (-x). They are symmetrical with respect to the y-axis. A line segment joining the points f (x) and f (-x) will be perfectly horizontal , shown by dotted blue line in the applet below.

Odd functions have the characteristic that f (x) = - f (-x). They are symmetrical with respect to the origin. A line segment joining the points f (x) and f (-x) always contains the origin , shown by dotted red line in the applet below. 

Some of the most common even functions are y = k , where k is a constant, y= x² , and y = cos(x) . Some of the most common odd functions are y = x³ and y = sin(x) . Some functions that are neither even nor odd include y = x - 4 , y = cos(x) + 1.

Formal tests for symmetry: 
1. y – axis : replace x with –x , produces an equivalent equation 
2. x - axis : replace y with – y , produces an equivalent equation 
3. origin : replace x with –x and y with –y , produces equivalent equation. 

In the following applet use check box to select between odd or even functions. Also you can enter any function in the input box to see whether it is odd or even by checking the symmetry about y-axis or origin.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com