Solving Geometry Right Triangles

Introduction :

In geometry, the triangles are made up of three line segments. When the two segments are perpendicular to each other (angle is 90 degree) then it is called as right triangle. The sizes of the angles and lengths of the sides are related to one another. If the triangle is a right triangle, we can be able to use simple trigonometric ratios to find the missing parts. For solving the right triangles in geometry we have to use Pythagoras Theorem to find missing sides and also the angles.

 

Solving geometry right triangles Using Pythagoras theorem:

 

The Pythagoras Theorem is developed by a Greek mathematician named Pythagoras, for solving the right triangle. It is also used to verify a triangle is Right angle triangle or not. According to the sum of square of two smaller sides is equal to the square of largest side. It is given by the formulae,

According to Pythagorean Theorem: c² = a² + b².

The largest side of a right angle triangle is always directly across from the 90 degree angle is the hypotenuse. The other two sides are called as legs.

We can also use the properties of sins, cosines, and tangents to solving the sides (a,b,c)of the triangles, that is, to find unknown parts in terms of known parts.

Sin A = `a/c` , Cos A = `b/c` , Tan A = `a/b`

 

Examples for solving geometry right triangles

 

Example1:

Solving third side for geometry right triangle, Given that a = 6, b = 8?

Solution:

Given

a = 6

b = 8

According to Pythagoras theorem,

   c² = a² + b² 

                                                    c² = 6² + 8²

c² = 36 + 64

                                                      c² = 100

                                                      c = `sqrt(100)`

                                                       c = 10 

 

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Example 2:

Solving one side a for geometry right triangle, given that A = 45°, b = 9?

Solution:

We know  that ,

Tan A = `a/b`                                               

                                                Tan 45 = `a/9 `

                                                1 = `a/9`       [Tan 45 = 1]

                                                9 = a

                                                a = 9.

The one side of geometry right triangle is 9.

Triangles In Geometry

Introduction :

Triangle is a three-sided polygon. ∆ABC denotes a triangle with vertices A, B, and C. There are several types of triangles in geometry based on the length of sides and the interior angles, some of them are equilateral triangle, scalene triangle, isosceles triangle, right triangle, oblique triangles, acute triangle, and obtuse triangle.

 

Types of Triangles in Geometry by Length of Sides:

 

• Equilateral Triangle:
  If all the three sides of the triangle are equal then the triangle is Geometry equilateral triangle. Here, all the three angles are also equal, i.e. 60°.

  

• Isosceles Triangle:
  If all the two sides of the triangle are equal then the triangle is geometry isosceles triangle. Here, all the three angles are also equal.

  

• Scalene triangle:
  In this case, all the sides and the angles are unequal.

 

Types of Triangles in Geometry by Interior Angles:

 

• Right-Angled Triangle:
  If any one of the triangles is 90, then the triangle is geometry right-angled triangle. The longest side in the triangle is hypotenuse. i.e. in ∆ABC, AC is hypotenuse.

  I am planning to write more post on Similar Triangles Proportions with example, Special Right Triangles Practice. Keep checking my blog.

  
  

  The Pythagorean Theorem states that, the sum of squares of length of two legs is equal to the hypotenuse of the right-angled triangle ABC. i.e. AC² = AB² + BC². E.g. Let us consider the right-angled triangle have the leg values AB = 12 and BC = 16, then the hypotenuse should be AC = 20.

• Oblique Triangles:
  Geometry Oblique triangle is a type of triangle where any of the interior angle is not equal to 90°. There are two cases under oblique triangle, they are,

          Case 1: Acute Triangle or acute – angled triangle
If all the angles in the triangle is less than 90°, then the triangle is acute-angled triangle.

          Case 2: Obtuse Triangles or obtuse-angled triangles
If any one of the angle in the triangle is more than 90°, then the triangle is said to be obtuse-angled triangle.

Triangles Solving Online

A triangle is the closed figure formed by three-line segments. Thus the triangle has mainly six parts such as three sides and three angles. The summation of angles of a triangle is 180°.The symbol Δ is used to represent the triangle. The summation of the measures of any two sides is always greater than the third side. Triangles are based on their sides and angles.Let us see triangles solving online in this topic.

 

Classification of triangles solving online:

 

Classification of triangles based on sides:

In this triangles solving online, triangles can be classified into three kinds based on the sides as follow:

• If all the three sides of a triangle are unequal then it is known as scalene triangle.
• If any two sides of a triangle are equal then it is known as isosceles triangle.
• If all the three sides of the triangle are equal then it is known as equilateral triangle.

Classification of triangles based on angles:

Based on the angles, triangles are divided into three types:

• If each angle of a triangle is an acute angle, that is, less than 90°, it is known as acute angled triangle.
• In a triangle, if any one angle is a right angle, that is equal to 90°, then the triangle is known as right angled triangle.
• In a triangle if any one angle is an obtuse angle, that is, greater than 90°, then the triangle is known as obtuse angled triangle.

 

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Examples to triangles solving online:

 

Ex: Can the following be the measure of the angles of a triangle?

a)    20°, 100°, 60°

b)   50°, 80°, 70°

Sol:

a)  The sum of the measure of the three angles is

20 + 100 + 60 = 180

Therefore 20°, 100°, 60° can be the measure of the angles of a triangle.

b)  The sum of the measure of the three angles is

50 + 80 + 70 = 200.

But the sum of the measure of the angles of a triangle is 180°.

Therefore 50°, 80°, 70° cannot be the measures of the angles of a triangle.

These are explains the concept of triangles solving online.

 

Hypotenuse Learning

Introduction:

    A hypotenuse is the longest side (side length) of a right-angled triangle (right triangle in American English), located opposite the right angle.

The side length of the hypotenuse of a right angle triangle can be found using the Pythagorean Theorem, which states that the square of the length of the hypotenuse(c) equals the sum of the squares of the lengths of the other two sides(a,b).

A right angle triangle is triangle with an angle of 90 degree. The sides a, b and c of such a given triangle satisfy the Pythagorean Theorem,

 c²= a²+b²

 

Properties and Formula

 

Properties:

In any right angle triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

• The length value of the hypotenuse equals the sum of the lengths of the orthographic projections of both cathetus.

• The square of the side length of a catheti equals the product of the lengths of its orthographic projection on the hypotenuse times the length of this.

                                        b² = a · m

                                        c² = a · n

• Also, the length of a catheti value b is the proportional mean between the lengths of its projection m and the hypotenuse a.

                                        a/b = b/m

                                        a/c = c/n

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Formula:   

Usually the length value of the hypotenuse is calculated using the square root function derived from the Pythagorean Theorem.

In mathematical notation;

c=√a²+b²

Here c is the value of hypotenuse.

a, b is the value of other two sides.

 

Examples

 

Example 1:

To find the value of hypotenuse with the other two side values are 6 cm, 8 cm.

Using the Pythagorean Theorem,

c²= a²+b²

c=√a² +b²

c=√6²+8²

c=√36+64

c=√100

c=√10*10

c=10

Example 2:

To find the value of hypotenuse with the value of x=3, y=4.

c=√x²+y²

c=√3²+4²

c=√9+16

c=√25

c=5

Lline Plots For Kids

• A line plot is defined as a graph that shows frequency of data along a number line. It is best to use a line plot when comparing fewer numbers.

• A line plot shows data on a number line with the symbol x or other marks to show frequency.

• In the applications of line plot, we have also created the scatter plot analysis. I we go in the wrong path or plotting the extra values or plotting the values in different place or leaving some values when we plotting make great difference in the diagrams.

line Plot for kids – Examples:

The count of cross marks above each score in the diagram represents the number of students who get the respective score. It is easy to calculate the percentage, average, mean, median, mode with their respective aspects. (This example easily understand for the kids)

line Plot for kids – solved example:

The following line plots represents the prices of different game books given below?

10, 30, 10, 10, 20, 30, 50, 40, 45, 50, 10, 30, 35, 30, 20, 40, 50

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Discuss whether Figure 1 or 2 is correct:

The Correct Answer is A

Solution:

Step 1: First arrange the data items from least to greatest.

$10, $10, $10, $10, $20, $20, $30, $30, $30, $30, $35, $40, $40, $45, $50, $50, $50

Step 2: Now group the data items that are the same.

4 game books cost $10

2 game books cost $20

4 game books cost $30

1 game book costs $35

2 game books cost $40

1 game book costs $45

3 game books cost $50

Step 3: Match the grouped data items with the figures shown. Figure-1 shows the data correctly.

We have also plot the line using different variability such as

• Number Line
• Data
• Frequency

Making a number line plot for kids:

•  To define the scale which we use, if the data is described as 100 the scale is increased by 10. So that we define the scale as 10 for 100

•  Next we have to draw the horizontal line to across the paper.

•  Break the grid as per the required scale.

•  Place the data values of x in the x-axis and y in the y-axis.

X      

                                   X      X

                                   X      X      X

                            X     X      X      X    X     X

0      1      2      3     4     5     6     7     8     9     10

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Similiar in Math Definition

In this article we are going discuss about similar triangle in math and its definition. If any two triangles corresponding sides will be at some ratio we can say that two triangles are similar. Here we are going to getting help on similar triangles. In similar triangles the angles are equal. The ratio between all the sides is equal. Using the ratio we can solve the sides of the triangles.
Example Problems for Similar in Math Definition:

We will discuss some example problems for similar triangles in math based on its definition. After viewing this article student can understand the concept.

Example 1 – Similar triangles in math:

Solve the side length of the triangle ABC. Triangle PQR and triangle ABC are similar.

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Similar triangle

Solution:

From the above the angles of the two triangles are equal. So the ratios between the triangles are equal. So we can find the sides of the triangles using this.

So

`(3sqrt(3))/(“AB”) = 3/(3sqrt(3))`

`(3sqrt(3))/(“AB”) = 1/(sqrt(3))`

`(3sqrt(3)) xx sqrt(3) = AB`

So AB = 9

Likewise we have to find the other side AC,

`(PR)/(AC) = (QR)/(BC)`

`6/(AC) = (3sqrt(3))/3`

AC = ` 6sqrt(3)`

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Example 2 – Similar Triangles in Math:

Solve for the variable y.

Similar triangle 2

Solution:

Here the above two triangles are similar. So

So `5/y = 6/12`

From this y = `60/6`

x = 10 cm.

Here the ratio between the sides is 2.

More example problems for similar triangles help:

Example 3 – Similar triangles in math:

Solve for the value x

Similar triangle 3

Solution:

These two triangles are similar. The ratios of triangles are equal.

`(BC)/(DE) = (AB)/(AD)`

`4/8 = y/ (y +4)`

4 (y + 4) = 8y

4y + 16 = 8y

4y = 16

y= 4.

These are some examples for similar triangles help. We can get the help on the similar triangles. . Normally if any two triangles are similar then ratio between any two sides will be equal. And included angles between these sides will be equal. It means the corresponding sides are in the same ratio and the corresponding angles are same. We can understand this from the above angles.

Different Systems of Measurement

Introduction :

In this we will see about different systems of measurement. Measurement is one of the most important tools in math. Without measurement we cannot measure any of our resultant data. Measurements are classified as many types. They are inches, meter, centimeters, kilograms, grams, miles, liters, pounds, and so on. Let us study about different systems of measurement.
Example Problems for Different Systems of Measurement:

Example problem 1: Add the followings: 3 T 1,088 lb + 4 T 241 lb

Solution:

We know that,

1 ton (T) = 2,000 pounds (lb)

Add the measurements. Add tons to tons and pounds to pounds.

3 T 1,088 lb + 4 T 241 lb = 7 T 1,329 lb

Answer: 3 T 1,088 lb + 4 T 241 lb = 7 T 1,329 lb

Example problem 2: At 1:00 AM, the temperature started dropping 2 degrees Celsius per hour until it reached -14 degrees Celsius at 4:00 AM. What was the temperature at 1:00 AM?

Solution:

Work backwards. The temperature dropped 2 degrees Celsius for 3 hours, so add 3 times:

-14°C + 2°C + 2°C + 2°C = -8°C

Therefore, the temperature was -8 degrees Celsius.

Answer: The temperature was -8 degrees Celsius.
Practice Problems for Different Systems of Measurement:

Practice problem 1: How many degrees Celsius (°C) is 113°F?

Practice problem 2: At 5:00 AM the temperature was 50°F, at 9:00 PM the temperature was 45°F, and at 10:00 PM the temperature was 43°F. When was it coolest?

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Practice problem 3: Multiply: 28 feet 2 inches times 2

Practice problem 4: Convert: 1/3 of a yard into feet.

Solutions for different systems of measurement:

Solution 1: 113°F = 45°C

Solution 2: 43 is the smallest of the three numbers, so 43°F is the coolest. So, it was coolest at 10:00 PM.

Solution 3: 28 feet 2 inches times 2 = 56 feet 4 inches

Solution 4: 1 feet

Area of Similar Shapes

Introduction :

In this article we shall discuss about the area of similar shapes, which are essential for learning geometrical shapes. Students are showing more interested by handling the shapes; here we are going to see how to measure the area of similar shape. The shapes like square, rectangle, circle and Triangle using their appropriate methods.
Area of Similar Shapes

Square shapes:

A square having the four sides with an equal length.

Area of square   = a2

square

Rectangle shape:

Rectangle has four sides and every angle is equal to 90degree, and it consists of length (l) and breadth (b).

Area of rectangular = lb

rectangle

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Triangle shape:

Triangle is closed polygon and it mentioned base (b) and height (h).

Area of triangle  =`1/2 bh`

triangle

Circle shape:

Circle is a curve generated by one point moving at a stable distance from a fixed point with the radius r.

Area of circle = `pi r^2`

circle
Examples for Area of Similar Shapes

Example 1:

Find the area of the square for similar shape in a rectangle 6cm length and 3 cm breadth.

rectangle

Solution:

Case (i):

we should take the AEFD

side(a)=3 cm

Area of square=a2

=(3)2

=9 cm2

Area of square = 9 cm2

Case(ii):

Now, take EBCF

side(a)=3 cm

Area of square=a2

=(3)2

=9

Area of square=9 cm2

Caes (i)=case(ii)

`therefore` the area of two square shapes are similar

Example 2:

Find the area of the triangle for similar shape in a square AB=4cm and BC=4 cm.

square

Solution:

Assume the triangle ABD,AB=4cm,AD=4cm

Area of triangle =`1/2 bh` <br>

=`1/2(4xx4) `

=`16/2`

=8 cm2

Area of triangle ABD=8 cm2

Now we assume the triangle BCD

Area of triangle  =`1/2 bh`

=1/2(4xx4)

=1/2(16)

=8 cm2

Area of triangle for BCD     =8 cm2

`therefore` the area of two triangle shapes are similar.

Practice examples for area of similar shapes:

1.Find the area of the triangle for similar shape in a  rectangle AB=8cm and BC=4cm.

Answer:16 cm2

2.Find the area of the triangle for similar shape in  a square  AB=6cm.

Answer:18 cm2

Square Triangular Numbers

Introduction :

In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinite number of square triangular numbers; the first few are 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025. Let us see about the topic which is square triangular number.

(Source: Wikipedia)
Explicit Formulas in Square Triangular Numbers

Write Nk for the kth square triangular number, and write down sk and tk for the regions of the equivalent square and triangle, so that

The series Nk, sk and tk are the OEIS sequences A001110, A001109, and A001108 in that order.

In 1778 Leonhard Euler determined the clear formula

Extra equivalent methods (obtained by expanding this formula) that may be suitable contains,
The equivalent explicit methods for sk and tk are
And
Recurrence Relations in Square Triangular Numbers

Here are return relations for the square triangular numbers, as fine as for the position of the square and triangle occupied. We has,

Nk = 34Nk – 1 – Nk – 2 + 2, with N0 = 0 and N1 = 1.

We contain

sk = 6sk – 1 – sk – 2, with s0 = 0 and s1 = 1;

tk = 6tk – 1 – tk – 2 + 2, with t0 = 0 and t1 = 1.

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Pell’s Equation in Square Triangular Numbers:

The sum of sentence square triangular numbers reduce to Pell’s formula in the follow way. Each triangular number is of the type t(t + 1)/2. Therefore we look for integers t, s such that

Among a bit of algebra this develop into,

(2t + 1)2 = 8s2 + 1,

And then letting x = 2t + 1 and y = 2s, we obtain the Diophantine equation
x2 – 2y2 = 1

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This is an instance of Pellӳ equation. This exacting equation is explain by the Pell numbers Pk as

And so all solutions are certain by

Here identity is told about the Pell integers, and these interpret into identities about the square triangular numbers.

Unit Circle Secant

Introduction:

In this article we shall see about terminal point on unit circle.Unit circle is a circle with the radius and centered at the origin in the  xy plane that means center point  is (0,0) .For any real number t, let P(x, y) be the point on the unit circle that is a distance t from (1, 0) in counterclockwise direction if t > 0 and in the clockwise direction if t < 0. We call the point P(x,y) the terminal point determined from t.Simply we say that the terminal point is the point reached after traveling t units around the circle.
Unit Circle

unit circle

A point is on the unit circle if it satisfies that  the equation of the circle, x2 + y2 = 1.

For example, (`3/5` ,`4/5` ) is on the unit circle since,

`(3/5)^(2)` +`(4/5)^(2)`

=1

The x-axes and y-axes divide the plane into four quadrants that is I, II, III, and IV, where each point (x, y) in the coordinate plane either lies on any one of the coordinate axes or any one of the four quadrants:

Quadrant I:

In this, both of the x and y are positive.
Quadrant II:

In this, x is negative and then y is positive.
Quadrant III:

In this,both of the x and y are negative.
Quadrant IV:

In this, x is positive and then y is negative.

Locating Terminal Points on the Unit Circle:

We all know the circumference of the circle is 2`pi`

Using that the following four rules are derived. we will use these rules for calculate the terminal points<br>

Separate the circle into 4 parts to calculate  the terminal points determined by the angles 0, `pi/2`,`pi` ,`3pi/2` and 2`pi `
Separate the circle into 8 parts to calculate  the terminal points determined by the angles `pi/4, “3pi/4` ,`5pi/4`,`7pi/4`
Separate the circle into 6 parts to calculate  the terminal points determined by the angles  `pi/3`,`2pi/3`,`4pi/3`,`5pi/3`
Separate the circle into 12 parts  to calculate the terminal points determined by the angles `pi/3`,`5pi/6`,`7pi/6`,`11pi/6`

Finding the Coordinates of terminal points in Quadrant I:

Terminal point p(x,y) determined  by  the angle `pi/4 `

In this angle,the point is at the intersection of the unit circle and the line y = x. Since

x2 + y2 = 1 and x = y,

it follows that 2×2 = 1

x=`sqrt(2)/2`              y=`sqrt(2)/2`

(  `sqrt(2)/2`, `sqrt(2)/2`

Terminal point determined by  the angle `pi/6`

In this angle,the  point (x; y) is the same distance from (x,-y) as it is from the point (0; 1). Therefore,

2y =`sqrt(x^(2)+(y-1)^(2))`  and so y=`1/2`

4y2=x2+(y-1)2

4.`(1/4)` =x2 +`1/4`

x2 =`3/4`

x = `(sqrt(3))/(2)`

( `sqrt(3)/2`,`1/2` )

Terminal point determined by the angle `pi/3`

In this angle the terminal point determined by  `pi/3` is the reflection of the terminal point determined by `pi/6` a cross the line y = x.so here the x and y values will be interchanged.

Example:

The point P is on the unit circle. Find the terminal point P (x, y) from the following given information.
The x-coordinate of P is `1/2` and P is in quadrant I.

solution:

`(1/2)^(2)` +y2 =1

y2 = 1-`(1/4)`

y2 = `3/4`

y =± `sqrt(3)/2`

y =`sqrt(3)/2` because  P is in quadrant I;

so, terminal point is (`1/2“sqrt(3)/2`)

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Finding the Coordinates of Terminal Points in any Quadrant:

We will consider that P(x , y) as a  terminal point determined by any value of t to find out the correct values of the terminal point.The following four rules will shows how to calculate the terminal point on unit circle in all of the quadrants of the circle:

(i) Place  the point P(x, y) on the unit circle.

(ii) Then calculate the  reference number t. This is the smallest distance along the unit circle between the
terminal point determined by t and the x-axis . The reference number  value is always positive.

(iii) calculate exact terminal point that is determined by t.

(iv) Modify  the signs of the coordinates according to the quadrant that contains the point  P(x,y).

Terminal points on unit circle in all quadrants

This diagram has been drawn using the above four rules.

unit circle with terminal point in all quadrants

From this diagram we can able to know the terminal points in all quadrant of the unit circle.