This is a session dedicated to three dimensional shapes namely 'SPHERE' & 'HEMISPHERE'. It's designed to explain the concept of surface area for both of these shapes using real life examples
Following are the subtopics covered here:
1. What is Sphere ?
2. Surface area of a sphere
3. Surface area of a hollow hemisphere
4. Surface area of a solid hemisphere
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
7.1 Introduction
7.2 Lines Of A Circle
7.3 Arcs
7.4 Inscribed Angles
7.5 Some Properties Of Tangents, Secants And Chords
7.6 Chords And Their Arcs
7.7 Segments Of Chords, Secants And Tangents
7.8 Lengths of Arcs And Areas Of Sectors
This is a session dedicated to three dimensional shapes namely 'SPHERE' & 'HEMISPHERE'. It's designed to explain the concept of surface area for both of these shapes using real life examples
Following are the subtopics covered here:
1. What is Sphere ?
2. Surface area of a sphere
3. Surface area of a hollow hemisphere
4. Surface area of a solid hemisphere
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
7.1 Introduction
7.2 Lines Of A Circle
7.3 Arcs
7.4 Inscribed Angles
7.5 Some Properties Of Tangents, Secants And Chords
7.6 Chords And Their Arcs
7.7 Segments Of Chords, Secants And Tangents
7.8 Lengths of Arcs And Areas Of Sectors
Ajie Ukpabi Asika. Funeral brochure June 2004Ed Keazor
Funeral brochure of Ajie Ukpabi Asika, showing tributes from friends, associates and family. Including President Olusegun Obasanjo, General Gowon, T.Y.Danjuma, Chu Okongwu, Ukwu I Ukwu, Jibril Aminu and many others.
These slides contain the pathagorean theorem and right trinagles. How to prove the oathagorean theorem and how to vind the area of triangles by the pathagorean theorem. There are some slides that explains that how the pathagorean theorem was discovrers. Some slides explain the pathagorean triple theorem and c^2=a^2 + b^2.
today we reviewed the Pythagorean Theorem and there was one sheet handed out to the B and D class and two handed out to the C class.
B and D make sure both sides are done, C make sure the front of each is done.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. Who was Pythagoras?
An ancient Greek thinker who was fond of poetry
and literature
He himself was not a geometer
His followers, dubbed the Pythagorean
Brotherhood, were a religious cult
Contrary to popular belief, Pythagoras was NOT
the founder of the Pythagorean Theorem
The Pythagorean Brotherhood founded the theorem
over 100 years after Pythagoras had died
3. What It Relates to:
Any and all right triangles
This means that this angle is a right angle
A right angle = 90º
Since there are 180º in a triangle, the other 2 angles must add
up to 90º
4. What Is It?
Let’s label the triangle by its angles ABC
If the triangle’s angles are ABC, then the sides
opposite those angles are a, b, and c,
respectively
AC
B
b
a
c
5. What Is It?
a and b are the sides, or legs, of the right
triangle
c is the hypotenuse, or the side opposite the
right angle, which is always the longest side of
the right triangle
In any triangle, the sums of any 2 sides is
greater than the length of the 3rd
side, so:
a + b > c
a + c > b
b + c > a
6. What Is It?
However, sometimes when we square the sides,
the sum of the squares is equal to the square of
the hypotenuse
a2
+ b2
= c2
This is the Pythagorean Theorem
The Pythagorean Theorem states:
In any right triangle, the sum of the squares of the 2 sides is
equal to the square of the hypotenuse.
Conversely, if the sum of the squares of the 2 sides is equal
to the square of the hypotenuse, then you have a right
triangle.
7. What Is It?
What does this mean?
It means that, if you have a right triangle, you
know that the squares of the 2 sides will
always add up to the square of the
hypotenuse.
It means that if you have a triangle in which
the squares of the 2 sides add up to the
square of the hypotenuse, you know you have
a right triangle.
8. What Is It?
Let’s look at some proofs:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
1 2 3
4 5 6
7 8 9
9. What Is It?
If you add up all of the little squares, i.e. 9
+ 16, you get 25. This is in accord with 32
+ 42
= 52
Here, we see that the lengths of the sides
are 3 and 4, and the length of the
hypotenuse is 5.
Using the Pythagorean Theorem, we get
32
+ 42
= 52
, or 9 + 16 = 25.
11. What Is It?
What is the area of the large blue square?
By multiplying the length by the width, we get (a + b)2
Simplify to a2
+ 2ab + b2
By adding up the areas of 4 small blue triangles and
the 1 small purple square, we get, 4(1/2)ab + c2
Simplify to 2ab + c2
If we set the areas equal to each other, we get a2
+
2ab + b2
= 2ab + c2
The 2ab’s cancel out, so we get a2
+ b2
= c2
Thus, the Pythagorean Theorem has been once
again proven.
12. What Is It?
The Pythagorean Theorem specifically refers to
squares of the sides, however any 2-
dimensional relation between the 2 sides and
the hypotenuse would work
If we drew circles on each side and the hypotenuse,
with the sides as the diameter of each respective
circle, then the areas of the circles on the 2 sides will
sum up to the area of the circle on the hypotenuse
Algebraically, this is seen as ka2
+ kb2
= kc2
since k
can be factored out, where k is some constant
13. Practice
Find the missing lengths:
a = 3, b = 4, c = ?
a = 7, b = ?, c = 25
a = ?, b = 12, c = 13
Are triangles with the following lengths right
triangles?
a = 7, b = 8, c = 9
a = 12, b = 16, c = 20
a = 11, b = 58, c = 61
14. Practice
To get from point A to point B you must avoid
walking through a pond. To avoid the pond, you
must walk 34 meters south and 41 meters east.
To the nearest meter, how many meters would
be saved if it were possible to walk through the
pond?
A baseball diamond is a square with sides of 90
feet. What is the shortest distance, to
thenearest tenth of a foot, between first base
and third base?
15. Practice
In a computer catalog, a computer monitor is
listed as being 19 inches. This distance is the
diagonal distance across the screen. If the
screen measures 10 inches in height, what is the
actual width of the screen to the nearest inch?
Oscar's dog house is shaped like a tent. The
slanted sides are both 5 feet long and the
bottom of the house is 6 feet across. What is
the height of his dog house, in feet, at its tallest
point?
16. Why Does It Matter?
The Pythagorean Theorem allows us to do many
things in real-life situations.
The professional fields in which it is useful are:
Civil Engineering
Construction
Astronomy
Physics
Particle Physics
Advanced Mathematics
Ancient Warfare
17. Why Does It Matter?
Civil Engineering:
Building bridges
Measuring distances across rivers in order to
determine the lengths of proposed bridges
Building foundations of skyscrapers
Construction
Measuring angles
Ensuring solid and level foundations
18. Why Does It Matter?
Astronomy
Measuring distances in a 3-dimensional
space
Calculating shadows cast by astronomical
bodies
Physics
Determining pressure in bridge construction
Understanding ramps, levers, and screws
19. Why Does It Matter?
Particle Physics
Calculating distances of particles in 3-dimensional space
Advanced Mathematics
Pythagorean Triples
Trigonometry and the Unit Circle
Vectors
Calculating distances between points on a Cartesian Plane
Ancient Warfare
The Ancient Romans used it to measure the distance that
catapults had to be from their target
20. Food for Thought
The distance formula for 2 points on a Cartesian Plane is derived
from the Pythagorean Theorem
The distance formula is d = √[(x2 – x1)2
+ (y2 – y1)2
]
This is simply a variation on c = √(a2
+ b2
), which is the Pythagorean
Theorem if you solve for c2
Pythagorean Triples are sets of 3 numbers that fit the criteria of a2
+
b2
= c2
Since any set of Pythagorean Triples can be multiplied by an infinite
amount of constants, there are an infinite amount of Pythagorean
Triples
If triangles with side lengths that corresponded to every Primitive
Pythagorean Triple (reduced by greatest common factor) were drawn on
a Cartesian Plane, we would end up with a unit circle, which is where
our Trigonometric functions come from