See the pages in numbers 7 to 10, which can be of help. From 5th post.

]]>Orthocenter indicates the center of all the right angles from the vertices to the opposite sides i.e., the altitudes. The term ortho means right and it is considered to be the intersection point of three altitudes drawn from the three vertices of a triangle. An orthocentre has significant importance in the study of the various properties of a triangle with respect to its other dimensions.

**Definition of an Orthocenter**

An orthocenter can be defined as the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides of a triangle. In a triangle, it is that point where all the three altitudes of a triangle intersect. The main three main aspects of an orthocenter are:

Triangle - A polygon with three vertices and three edges.

Altitude - The altitude of a triangle is that line that passes through its vertex and is perpendicular to the opposite side.

Hence, a triangle can have three altitudes, one from each vertex.

Vertex - The point where two or more lines meet is called a vertex.

Look at the image below, △ABC is a triangle, △ABC has three altitudes, namely, AE, BF, and, CD, △ABC has three vertices, namely, A, B, and, C, and the intersection point H is the orthocenter.

**Properties of an Orthocenter**

The properties of an orthocenter vary depending on the type of triangle such as the Isosceles triangle, Scalene triangle, right-angle triangle, etc. For some triangles, the orthocenter need not lie inside the triangle but can be placed outside. For instance, for an equilateral triangle, the orthocenter is the centroid.

**How to Construct an Orthocenter?**

To construct the orthocenter for a triangle geometrically, we have to do the following:

Find the perpendicular from any two vertices to the opposite sides.

To draw the perpendicular or the altitude, use vertex C as the center and radius equal to the side BC. Draw arcs on the opposite sides AB and AC.

Draw intersecting arcs from B and D, at F. Join CF.

Similarly, draw intersecting arcs from points C and E, at G. Join BG.

CF and BG are altitudes or perpendiculars for the sides AB and AC respectively.

The intersection point of any two altitudes of a triangle gives the orthocenter.

Thus, find the point of intersection of the two altitudes.

At that point, H is referred to as the orthocenter of the triangle.

PA, QB, RC are the perpendicular lines drawn from the three vertices

`[P[(x)_1, (y)_1], Q[(x)_2, (y)_2], \ and \ R[(x)_3, (y)_3] \ respectively \ of \ the \ \triangle \ PQR`

gives

H ( x, y) is the intersection point of the three altitudes of the triangle.

Calculate the slope of the sides of the triangle using the formula:

`m(slope) = \frac{(y_2 - y_1)}{(x_2 - x_1)}`

gives

Let slope of PR be given by mPR.

Hence,

`mPR = \frac{(y_3 - y_1)}{(x_3 - x_1)}`

gives

Similarly,

`mQR = \frac{(y_3 - y_2)}{(x_3 - x_2)}`

gives

Step 2 The slope of the altitudes of the △PQR will be perpendicular to the slope of the sides of the triangle.

The generalized equation thus formed by using arbitrary points (x) and (y) is:

`mPA = \frac{(y - y_1)}{(x - x_1)}`

gives

`mQB = \frac{(y - y_2)}{(x - x_2)}`

gives

Thus, solving the two equations for any given values the orthocenter of a triangle can be calculated.

]]>The centroid of a triangle is a point of concurrency of the medians of a triangle. Before understanding the point of concurrency, let us discuss the medians of a triangle. Medians are the line segments that are drawn from the vertex to the mid-point of the opposite side of the vertex. Each median of a triangle divides the triangle into two smaller triangles that have equal areas. The point of intersection of the medians of a triangle is known as centroid. The centroid always lies inside a triangle, unlike other points of concurrencies of a triangle.

**What is Centroid of a Triangle?**

The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrencies of a triangle. The medians of a triangle are constructed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle. ntroid of a triangle

**Properties of the Centroid of Triangle**

The following points show the properties of the centroid of a triangle which are very helpful to distinguish the centroid from all the other points of concurrencies.

* The centroid is also known as the geometric center of the object.

* The centroid of a triangle is the point of intersection of all the three medians of a triangle.

* The medians are divided into a 2:1 ratio by the centroid.

* The centroid of a triangle is always within a triangle.

**Centroid of Triangle Formula**

The centroid of a triangle formula is used to find the centroid of a triangle uses the coordinates of the vertices of a triangle. The coordinates of the centroid of a triangle can only be calculated if we know the coordinates of the vertices of the triangle. The formula for the centroid of the triangle is:

`C(x,y) = (x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3`

gives

or

`C(x,y) = \left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3}\right)`

gives

where, x1, x2, and x3 are the 'x-coordinates' of the vertices of the triangle; and y1, y2, and y3 are the 'y-coordinates of the vertices of the triangle.

**Important Notes on Centroid of Triangle**

* The centroid of a triangle is the point of intersection of the medians of a triangle.

* It always lies inside the triangle.

* Centroid divides the medians in the ratio 2:1.

To indent subsequent lines of a paragraph, use the TeX command \hangindent . (While the default behaviour is to apply the hanging indent after the first line, this may be changed with the \hangafter command.) An example follows. \hangindent=0.7cm This paragraph has an extra indentation at the left.

This link helps:

thanks!

This link helps:

]]>JaneFairfax wrote:

You can even specify exactly how much spacing you want. For example, if you want exactly 30 mm of white space,

`\hspace{30mm}`

Thus:

but text isnt indented

is there a way to indent?

\quad doesnt work

\text{ } doesnt work

p.s. nehushtn posted here: https://www.mathisfunforum.com/viewtopic.php?pid=291982#p291982

Nehushtan wrote:

that is so frikkin awesome!

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

**Incenter Formula**

To calculate the incenter of a triangle with 3 cordinates, we can use the incenter formula. Let us learn about the formula. Consider the coordinates of incenter of the triangle ABC with coordinates of the vertices,

`A((x)_1, (y)_1), B((x)_2, (y)_2), C((x)_3, (y)_3)`

gives

and sides a, b, c are:`\left(\dfrac{ax_1 + bx_2 + cx_3}{a + b+ c}, \dfrac{ay_1 + by_2 + cy_3}{a + b+ c}\right)`

gives

What is the distributive property?

Solution:

The distributive property is a property of multiplication used in addition and subtraction.

Distributive property states that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

It also applies for difference of numbers being multiplied to a number.

It is given as:

`a \times (b + c) = a \times b + a \times c`

gives

.`a \times (b - c) = a \times b - a \times c`

gives

Example 1 :

`3 \times ( 2 + 5 ) = 3 \times 2 + 3 \times 5`

gives

`LHS = 3 \times ( 2 + 5 ) = 3 \times 7 = 21`

gives

`RHS = 3 \times 2 + 3 \times 5 = 6 + 15 = 21`

gives

Thus, LHS = RHS

Example 2 :

`5 \times ( 4 - 1) = 5 \times 4 - 5 \times 1`

gives

.`LHS = 5 \times ( 4 - 1) = 5 \times 3 = 15`

gives

`RHS = 5 \times 4 - 5 \times 1 = 20 - 5 = 15`

gives

Hence, LHS = RHS

Thus, we have understood the usage of the distributive property by looking into the examples.

What is the distributive property?

Summary:

Distributive property states that two or more terms in addition or subtraction, when multiplied with a number are equal to the addition or subtraction of the product of each of the terms with that number.

]]>The associative property, or the associative law in mathematics, states that while adding or multiplying numbers, the way in which numbers are grouped by brackets (parentheses), does not affect their sum or product. The associative property is applicable to addition and multiplication. Let us learn more about the associative property, along with some associative property examples.

**What is the Associative Property?**

According to the Associative property, when 3 or more numbers are added or multiplied, the result (sum or the product) remains the same even if the numbers are grouped in a different way. Here, grouping is done with the help of brackets. This can be expressed as,

`a \times (b \times c) = (a \times b) \times c`

gives

`a + (b + c) = (a + b) + c`

gives

.**Associative Property Definition**

The associative law which applies only to addition and multiplication states that the sum or the product of any 3 or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.

The formula for the associative property of addition and multiplication is expressed as:

**Formula for the Associative law of addition and multiplication**

Let us discuss in detail the associative property of addition and multiplication with examples.

**Associative Property of Addition**

According to the associative property of addition, the sum of three or more numbers remains the same irrespective of the way the numbers are grouped. Suppose we have three numbers: a, b, and c. For these, the associative property of addition will be expressed with the following formula:

Associative Property of Addition Formula:

`(A + B) + C = A + (B + C)`

gives

.Let us understand this with the help of an example.

Example: (1 + 7) + 3 = 1 + (7 + 3) = 11. If we solve the left-hand side, we get, 8 + 3 = 11. Now, if we solve the right-hand side, we get, 1 + 10 = 11. Hence, we can see that the sum remains the same even when the numbers are grouped in a different way.

**Associative Property of Multiplication**

The associative property of multiplication states that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property of multiplication can be expressed with the help of the following formula:

Associative Law of Multiplication Formula

`(A \times B) \times C = A \times (B \times C)`

gives

Let us understand this with the following example.

Example:

`(1 \times 7) \times 3 = 1 \times (7 \times 3) = 21`

gives

When we solve the left-hand side, we get

`7 × 3 = 21`

gives

Now, when we solve the right-hand side, we get

`1 × 21 = 21`

gives

.Therefore, it can be seen that the product of the numbers remains the same irrespective of the different grouping of numbers.

**Associative Property of Subtraction**

The associative property does not work with subtraction. This means if we try to apply the associative law to subtraction, it will not work. For example, (7 - 1) - 3 is not equal to 7 - (1 - 3). If we solve the left-hand side, we get, 6 - 3 = 3. Now, if we solve the right-hand side, we get, 7 - (-2) = 9. Hence, we can see there is no associative property of subtraction.

**Verification of Associative Law**

Let us try to justify how and why the associative property is only valid for addition and multiplication operations. We will apply the associative law individually on the four basic operations.

For Addition: The associative law in Maths for addition is expressed as (A + B) + C = A + (B + C). So, let us substitute this formula with numbers to verify it. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. Therefore, the associative property is applicable to addition.

For Subtraction: Let us try the associative property formula in subtraction. This can be expressed as (A - B) - C ≠ A - (B - C). Now, let us verify this formula by substituting numbers in this. For example,

`(1 - 4) - 2 \neq 1 - (4 - 2)`

gives

i.e., -5 ≠ -1. Therefore, we say that the associative property is not applicable to subtraction.For Multiplication: The associative law for multiplication is given as

`(A \times B) \times C = A \times (B \times C)`

gives

For example,`(1 \times 4) \times 2 = 1 \times (4 \times 2) = 8`

gives

Therefore, we can say that the associative property is applicable to multiplication.

For Division: Now, let us try the associative property formula for division. This can be expressed as

`(A \div B) \div C \neq A \div (B \div C)`

gives

For example,

`(9 \div 3) \div 2 \neq 9 \div (3 \div 2) = 3/2 \neq 6`

gives

.Therefore, we can see that the associative property is not applicable to division.]]>

The commutative property applies to the arithmetic operations of addition and multiplication. It means that changing the order or position of two numbers while adding or multiplying them does not change the end result. For example, 4 + 5 gives 9, and 5 + 4 also gives 9. The order of two numbers being added does not affect the sum. The same concept applies to multiplication too. The commutative property does not hold for subtraction and division, as the end results are completely different after changing the order of numbers.

**What is Commutative Property?**

The word 'commutative' originates from the word 'commute', which means to move around. Hence, the commutative property deals with moving the numbers around. So mathematically, if changing the order of the operands does not change the result of the arithmetic operation then that particular arithmetic operation is commutative. Let us discuss the commutative property of addition and multiplication.

**Commutative Property Formula**

For any two numbers, A and B, the formula of the commutative property of numbers is expressed as follows.

`A + B = B + A`

gives

.`A \times B = B \times A`

gives

`A - B \neq B - A`

gives

`A \div B \neq B \div A`

gives

.The commutative property formula states that the change in the order of two numbers while adding and multiplying them does not affect the result. However, while subtracting and dividing any two real numbers, the order of numbers are important and hence it can't be changed.

**Commutative Property of Addition**

The commutative property of addition says that changing the order of the addends does not change the value of the sum. If 'A' and 'B' are two numbers, then the commutative property of addition of numbers can be represented as shown in the figure given below.

**Commutative Property of Addition Formula**

Let us take an example of the commutative property of addition and understand the application of the above formula.

Example: Let us check the Commutative property by adding 10 and 13.

Let us add the given numbers 10 and 13. So, 10 + 13 = 23 and 13 + 10 = 23. Therefore, 10 + 13 = 13 + 10 which proves the commutative property of addition.

**Commutative Property of Multiplication**

The commutative property of multiplication says that the order in which we multiply two numbers does not change the final product. The figure given below represents the commutative property of the multiplication of two numbers.

**Commutative Property of Multiplication**

If 4 and 6 are the numbers, then

`4 \times 6 = 24, \ and \ 6 \times 4`

gives

is also equal to 24. Thus`4 \times 6 = 6 \times 4`

gives

. Therefore, the commutative property holds true for the multiplication of numbers.Note: The commutative property does not hold for subtraction and division operations. Let us take the example of numbers 6 and 2.

`6 - 2 = 4, \ but \ 2 - 6 = -4. \ Thus,\ 6 - 2 \neq 2 - 6`

gives

.`6 \div 2 = 3, \ but \ 2 \div 6 = 1/3. \ Thus, 6 \div 2 \neq 2 \div 6`

gives

**Commutative Property of Subtraction**

The commutative property is not applicable to subtraction. The commutative law only applies to addition and multiplication. Let us see why it does not apply on subtraction. For example,

`8 - 5 = 3, \ but \ 5 - 8 = -3. \ Thus, 8 - 5 \neq 5 - 8`

gives

Rectangular Hyperbola is a hyperbola having the transverse axis and the conjugate of 2a units and conjugate axis of 2b units of equal length. The eccentricity of a rectangular hyperbola is

`\sqrt{2}`

gives

, and the equation of a rectangular hyperbola is`x^2 - y^2 = a^2`

gives

**What Is A Rectangular Hyperbola?**

A rectangular hyperbola is a hyperbola having the transverse axis and the conjugate axis of equal length. The arcs of a rectangular hyperbola is the same as the arc of a circle. For a rectangular hyperbola having the transverse axis of length 2a and the conjugate axis of length 2b, we have 2a = 2b, or a = b. The general equation of a rectangular hyperbola is

`x^2 - y^2 = a^2.`

gives

The equation of asymptotes of a rectangular hyperbola is

`y = \pm \ or \ x \ or \ x^2 - y^2 = 0`

gives

The axes or the asymptotes of the rectangular hyperbola are perpendicular to each other. The rectangular hyperbola is related to a hyperbola in a similar form as the circle is related to an ellipse. The eccentricity of a rectangular hyperbola is

`\sqrt{2}`

gives

The graph of the equation y = 1/x is similar to the graph of a rectangular hyperbola.**Properties of Rectangular Hyperbola**

The rectangular hyperbola is similar to a regular hyperbola, and the only difference is the different lengths of the transverse axis and conjugate axis in a hyperbola, and these lengths are equal in a rectangular hyperbola The following are some of the important properties of a rectangular hyperbola.

* The eccentricity of a rectangular hyperbola is equal to

`\sqrt{2}`

gives

* The transverse axis and the conjugate axis in a rectangular hyperbola is of equal length.

* The asymptotoes of a rectangular hyperbola is

`y = \pm \ x \ or \ x^2 - y^2 = 0`

gives

* The asymptotes of a rectangular hyperbola are perpendicular to each other.

* The conjugate of a rectangular hyperbola

`x^2 - y^2 = a^2`

gives

is also a rectangular hyperbola`x^2 - y^2 = -a^2`

gives

* The parametric form of representation of a rectangular hyperbola has the coordinates

`x = aSec\theta, y = aTan\theta`

gives

]]>In mathematics, a hyperbola is an important conic section formed by the intersection of the double cone by a plane surface, but not necessarily at the center. A hyperbola is symmetric along the conjugate axis, and shares many similarities with the ellipse. Concepts like foci, directrix, latus rectum, eccentricity, apply to a hyperbola. A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of sub-atomic particles, etc.

Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, and standard forms of hyperbola using the solved examples.

**What is Hyperbola?**

A hyperbola, a type of smooth curve lying in a plane, has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y) on the hyperbola and for two foci F, F', the locus of the hyperbola is PF - PF' = 2a.

**Hyperbola Definition**

A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

**Parts of a Hyperbola**

Let us check through a few important terms relating to the different parameters of a hyperbola.

Foci of hyperbola: The hyperbola has two foci and their coordinates are F(c, o), and F'(-c, 0).

Center of Hyperbola: The midpoint of the line joining the two foci is called the center of the hyperbola.

Major Axis: The length of the major axis of the hyperbola is 2a units.

Minor Axis: The length of the minor axis of the hyperbola is 2b units.

Vertices: The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (-a, 0).

Latus Rectum of Hyperbola: The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is

`2b^2/a`

gives

.Transverse Axis: The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola.

Conjugate Axis: The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

Eccentricity of Hyperbola: (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the hyperbola, and the distance of the vertex from the center of the hyperbola. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a.

**Hyperbola Equation**

The below equation represents the general equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.

`\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1`

gives

.Let us understand the standard form of the hyperbola equation and its derivation in detail in the following sections.

**Standard Equation of Hyperbola**

There are two standard equations of the Hyperbola. These equations are based on the transverse axis and the conjugate axis of each of the hyperbola. The standard equation of the hyperbola is

`\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1`

gives

.has the transverse axis as the x-axis and the conjugate axis is the y-axis. Further, another standard equation of the hyperbola is

`\dfrac{y^2}{a^2} - \drac{x^2}{b^2} = 1`

gives

and it has the transverse axis as the y-axis and its conjugate axis is the x-axis.

**Eccentricity(e) of hyperbola formula:**

`e = \sqrt{1 + \dfrac{b^2}{a^2}}`

gives

.]]>An annulus is an inner region between two concentric circles i.e. two or more circles sharing the same center point. The annulus is shaped like a ring and has many applications in mathematics that we will be learning in this article. Some of the real-life examples are a doughnut, finger rings. etc. Let us learn more about the shape of the annulus and solve a few examples to understand the concept better.

**Annulus Definition**

An annulus is a two-dimensional flat figure shaped in a circular form which is constructed by two concentric circles. The region or the area formed in between these two concentric circles is called the annulus. Since it is a flat figure in a circular form, the edges are two circles with the same center. It is considered a circular disk having a circular hole in the middle.

**Annulus Meaning**

The word annulus is derived from a Latin word, 'annuli', meaning little rings. The shape of the annulus is flat and circular with a hole in between, much like a throw ring or a circular disc. Look at the image below showing two circles i.e. one small circle also called an inner circle and a big circle also called the outer circle. The point which is marked as red is the center of both circles. The shaded colored area, between the boundary of these two circles, is known as an annulus.

**Area of the Annulus**

The annulus area is the area of the ring-shaped space i.e. the enclosed region between the two concentric circles. To calculate the area of the annulus, we need the area of both the inner circle and the outer circle. The dimensions of an annulus are defined by the two radii R, and r, which are the radii of the outer ring and the inner ring respectively. Once the measurements of both the radii are known, we can calculate the area by subtracting the area of the small circle from the big circle. Hence, the formula used for finding the area of the annulus is:

Area of Outer Circle =

`\pi{R}^2`

gives

.Area of Inner Circle =

`\pi{r}^2`

gives

.Area of Annulus = Area of Outer Circle – Area of Inner Circle

Therefore, Area of Annulus =

`\pi(R^2 - r^2)`

gives

square units, or it can be written as

`\pi(R + r)(R - r)`

gives

square units, where R is the radius of the outer circle, r is the radius of the inner circle, and π(pi) is approximately 3.142.

The area of the outer (bigger) circle - the area of the inner (smaller) circle = the area of the annulus.

**Annulus Perimeter**

The perimeter is the distance around the 2D shape. Since the annulus is a flat circular shape constructed by two concentric circles, it can also be considered as a ring. Therefore, an open ring can be considered as the topological equivalent of a cylinder and a punctured plane. Similar to the area, to find the perimeter of the annulus we need to consider both the inner circle and the outer circle. So, the perimeter of the ring or annulus is equal to the sum of the radii of the large and small circles multiplied by 2π. The formula for finding the perimeter is:

Perimeter of Annulus (P) =

`2\pi(R + r)`

gives

units, where R is the radius of the outer circle, r is the radius of the inner circle, and`\pi`

gives

(pi) is approximately 3.142.]]>