This document discusses formulas for finding measurements of triangles, quadrilaterals, and other geometric shapes. It includes examples of using formulas to find heights, bases, areas, and perimeters given certain measurements. Formulas are provided for triangles, parallelograms, rectangles, squares, trapezoids, rhombuses, kites, and applying these formulas to problems involving geometric grids. Lesson objectives and examples with step-by-step solutions demonstrate how to use the formulas to calculate missing values.
The Earth is not flat; but it's not round either (Geography on Boost.Geometry)Vissarion Fisikopoulos
What is a great circle, a loxodrome or a geodesic? What are the differences between them and which one is more suitable for each GIS application? This talk addresses this kind of questions and how geography is implemented in Boost.Geometry. The library that is currently being used to provide GIS support to MySQL.
Following up the introductory talk on Boost.Geometry we discuss the algorithmic, the implementation as well as the user perspective of the development of geography in Boost.Geometry. We define basic geometric objects such as geodesics, and the modeling of the Earth as a sphere or ellipsoid. We try to understand the effect of different Earth models to the accuracy and speed of fundamental geometric algorithms (such as length, area, intersection etc.) by showing particular examples. Finally, we are having a look towards the future of geography in Boost.Geometry.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
The Earth is not flat; but it's not round either (Geography on Boost.Geometry)Vissarion Fisikopoulos
What is a great circle, a loxodrome or a geodesic? What are the differences between them and which one is more suitable for each GIS application? This talk addresses this kind of questions and how geography is implemented in Boost.Geometry. The library that is currently being used to provide GIS support to MySQL.
Following up the introductory talk on Boost.Geometry we discuss the algorithmic, the implementation as well as the user perspective of the development of geography in Boost.Geometry. We define basic geometric objects such as geodesics, and the modeling of the Earth as a sphere or ellipsoid. We try to understand the effect of different Earth models to the accuracy and speed of fundamental geometric algorithms (such as length, area, intersection etc.) by showing particular examples. Finally, we are having a look towards the future of geography in Boost.Geometry.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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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.
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Macroeconomics- Movie Location
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Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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.
1. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals9-1
Developing Formulas for
Triangles and Quadrilaterals
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
2. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Warm Up
Find the unknown side length in each
right triangle with legs a and b and
hypotenuse c.
1. a = 20, b = 21
2. b = 21, c = 35
3. a = 20, c = 52
c = 29
a = 28
b = 48
3. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Develop and apply the formulas for the
areas of triangles and special
quadrilaterals.
Solve problems involving perimeters
and areas of triangles and special
quadrilaterals.
Objectives
4. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
A tangram is an ancient Chinese puzzle made
from a square. The pieces can be rearranged to
form many different shapes. The area of a figure
made with all the pieces is the sum of the areas of
the pieces.
5. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Recall that a rectangle with base b and height h has
an area of A = bh.
You can use the Area Addition Postulate to see that
a parallelogram has the same area as a rectangle
with the same base and height.
6. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Remember that rectangles and squares are also
parallelograms. The area of a square with side s is
A = s2
, and the perimeter is P = 4s.
7. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
The height of a parallelogram is measured
along a segment perpendicular to a line
containing the base.
Remember!
8. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
The perimeter of a rectangle with base b
and height h is P = 2b + 2h or
P = 2 (b + h).
Remember!
9. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Find the area of the parallelogram.
Example 1A: Finding Measurements of Parallelograms
Step 1 Use the Pythagorean Theorem to
find the height h.
Step 2 Use h to find the area of the parallelogram.
Simplify.
Substitute 11 for b and 16 for h.
Area of a parallelogram
302
+ h2
= 342
h = 16
A = bh
A = 11(16)
A = 176 mm2
10. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 1B: Finding Measurements of Parallelograms
Find the height of a rectangle in which b = 3 in.
and A = (6x² + 24x – 6) in2
.
Sym. Prop. of =
Divide both sides by 3.
Factor 3 out of the expression
for A.
Substitute 6x2
+ 24x – 6
for A and 3 for b.
Area of a rectangleA = bh
6x2
+ 24x – 6 = 3h
3(2x2
+ 8x – 2) = 3h
2x2
+ 8x – 2 = h
h = (2x2
+ 8x – 2) in.
11. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 1C: Finding Measurements of Parallelograms
Find the perimeter of the rectangle, in which
A = (79.8x2
– 42) cm2
Step 1 Use the area and the height to
find the base.
Substitute 79.8x2
– 42 for
A and 21 for h.
Divide both sides by 21.
Area of a rectangleA = bh
79.8x2
– 42 = b(21)
3.8x2
– 2 = b
12. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 1C Continued
Step 2 Use the base and the height to find the
perimeter.
Simplify.
Perimeter of a rectangle
Substitute 3.8x2
– 2 for b
and 21 for h.
P = 2b + 2h
P = 2(3.8x2
– 2) + 2(21)
P = (7.6x2
+ 38) cm
13. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Check It Out! Example 1
A = bh
Find the base of the parallelogram in which
h = 56 yd and A = 28 yd2
.
28 = b(56)
56 56
b = 0.5 yd
Area of a parallelogram
Substitute.
Simplify.
15. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Find the area of a trapezoid in which b1 = 8 in.,
b2 = 5 in., and h = 6.2 in.
Example 2A: Finding Measurements of Triangles and
Trapezoids
Simplify.
Area of a trapezoid
Substitute 8 for b1, 5 for b2,
and 6.2 for h.
A = 40.3 in2
16. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 2B: Finding Measurements of Triangles and
Trapezoids
Find the base of the triangle, in which
A = (15x2
) cm2
.
Sym. Prop. of =
Divide both sides by x.
Substitute 15x2
for A and
5x for h.
Area of a triangle
6x = b
b = 6x cm
Multiply both sides by
17. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 2C: Finding Measurements of Triangles and
Trapezoids
Find b2 of the trapezoid,
in which A = 231 mm2
.
2
11
Multiply both sides by .
Sym. Prop. of =
Subtract 23 from both sides.
b1
Substitute 231 for A, 23 for ,
and 11 for h.
Area of a trapezoid
42 = 23 + b2
19 = b2
b2 = 19 mm
18. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Check It Out! Example 2
Find the area of the triangle.
Find b.
A = 96 m2
Substitute 16 for b and
12 for h.
Area of a triangle
Simplify.
19. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
The diagonals of a rhombus or kite are
perpendicular, and the diagonals of a
rhombus bisect each other.
Remember!
21. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 3A: Finding Measurements of Rhombuses
and Kites
Find d2 of a kite in which d1 = 14 in. and
A = 238 in2
.
Area of a kite
Substitute 238 for A and 14 for d1.
Solve for d2.
Sym. Prop. of =
34 = d2
d2 = 34
22. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 3B: Finding Measurements of Rhombuses
and Kites
Find the area of a rhombus.
.
Substitute (8x+7) for d1
and (14x-6) for d2.
Multiply the binomials
(FOIL).
Distrib. Prop.
Area of a rhombus
23. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 3C: Finding Measurements of Rhombuses
and Kites
Find the area of the kite
Step 1 The diagonals d1 and
d2 form four right triangles.
Use the Pythagorean Theorem
to find x and y.
282
+ y2
= 352
y2
= 441
y = 21
212
+ x2
= 292
x2
= 400
x = 20
24. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Step 2 Use d1 and d2 to find the area.
d1 is equal to x + 28, which is 48.
Half of d2 is equal to 21, so d2 is equal to 42.
A = 1008 in2
Area of kite
Substitute 48 for d1
and 42 for d2.
Simplify.
Example 3C Continued
25. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Check It Out! Example 3
Find d2 of a rhombus in which d1 = 3x m
and A = 12xy m2
.
d2 = 8y m
Formula for area of a rhombus
Substitute.
Simplify.
26. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 4: Games Application
The tile design shown is a rectangle with a
base of 4 in. and a height of 2 in. Use the grid
to find the perimeter and area of the leftmost
shaded parallelogram.
Perimeter:
Two sides of the parallelogram
are vertical and the other two
sides are diagonals of a square
of the grid. Each grid square has a side length of 1 in.,
so the diagonal is The perimeter of the leftmost
shaded parallelogram is P = 2(1)+2( ) = (2 +
2 ) in.
27. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Example 4 Continued
The tile design shown is a rectangle with a
base of 4 in. and a height of 2 in. Use the grid
to find the perimeter and area of the leftmost
shaded parallelogram.
Area:
The base and height of the
leftmost shaded parallelogram
each measure 1 in., so the area
is A = bh = (1)(1) = 1 in2
.
28. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Check It Out! Example 4
In the tangram, find the perimeter and area of
the large green triangle. Each grid square has
a side length of 1 cm.
The area is A = 4cm2
.
The perimeter is
P = (4 + 4 ) cm.
29. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Lesson Quiz: Part I
Find each measurement.
1. the height of the parallelogram, in which
A = 182x2
mm2
h = 9.1x mm
2. the perimeter of a rectangle in which h = 8 in.
and A = 28x in2
P = (16 + 7x) in.
30. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Lesson Quiz: Part II
3. the area of the trapezoid
A = 16.8x ft2
4. the base of a triangle in which
h = 8 cm and A = (12x + 8) cm2
b = (3x + 2) cm
5. the area of the rhombus
A = 1080 m2
31. Holt Geometry
9-1
Developing Formulas for
Triangles and Quadrilaterals
Lesson Quiz: Part III
6. The wallpaper pattern shown is a rectangle with a
base of 4 in. and a height of 3 in. Use the grid to
find the area of the shaded kite.
A = 3 in2
