This document provides instruction on similarity in right triangles. It begins with examples of identifying similar right triangles and finding geometric means. Geometric mean is defined as the positive square root of the product of two numbers. The document then shows how geometric means can be used to find unknown side lengths in right triangles using proportions. Examples demonstrate applications to measurement problems. The lesson concludes with a quiz reviewing key concepts like writing similarity statements and using proportions with geometric means to solve for missing side lengths in right triangles.
This powerpoint presentation is an introduction for the topic TRIANGLE CONGRUENCE. This topic is in Grade 8 Mathematics. I hope that you will learn something from this sides.
This powerpoint presentation is an introduction for the topic TRIANGLE CONGRUENCE. This topic is in Grade 8 Mathematics. I hope that you will learn something from this sides.
This will help you on how to solve quadratic equations by factoring.
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You've seen that many quantities are related to each other. However, not all of them are directly related. Now you will explore quantities that vary inversely. In inverse variation, one quantity decreases as the other increases.
This will help you on how to solve quadratic equations by factoring.
For more instructional resources, CLICK me here!
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You've seen that many quantities are related to each other. However, not all of them are directly related. Now you will explore quantities that vary inversely. In inverse variation, one quantity decreases as the other increases.
Maths (CLASS 10) Chapter Triangles PPT
thales theorem
similar triangles
phyathagoras theorem ,etc
In this ppt all theorem are proved solution are gven
there are videos also
all topic cover
GREKing: The most repeated type of quants problem.Rahul Singh
GREKing the most repeated types of quants problems which can be scoring for the initial section.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
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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A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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!
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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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.
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• 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.
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• 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
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.
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.
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1. 8-1 Similarity in Right Triangles
8-1 Similarity in Right Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
Holt Geometry
2. 8-1 Similarity in Right Triangles
Warm Up
1. Write a similarity statement
comparing the two triangles.
∆ADB ~ ∆EDC
Simplify.
2.
3.
Solve each equation.
4.
Holt Geometry
5. 2x2 = 50
±5
3. 8-1 Similarity in Right Triangles
Objectives
Use geometric mean to find segment lengths
in right triangles.
Apply similarity relationships in right triangles
to solve problems.
Holt Geometry
4. 8-1 Similarity in Right Triangles
Vocabulary
geometric mean
Holt Geometry
5. 8-1 Similarity in Right Triangles
In a right triangle, an altitude drawn from the vertex of the
right angle to the hypotenuse forms two right triangles.
Holt Geometry
7. 8-1 Similarity in Right Triangles
Example 1: Identifying Similar Right Triangles
Write a similarity statement
comparing the three triangles.
Sketch the three right triangles with the angles of
the triangles in corresponding positions.
W
Z
By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.
Holt Geometry
8. 8-1 Similarity in Right Triangles
Check It Out! Example 1
Write a similarity statement comparing
the three triangles.
Sketch the three right triangles with the
angles of the triangles in corresponding
positions.
By Theorem 8-1-1, ∆LJK ~ ∆JMK ~ ∆LMJ.
Holt Geometry
9. 8-1 Similarity in Right Triangles
Consider the proportion
. In this case, the
means of the proportion are the same number, and
that number is the geometric mean of the extremes.
The geometric mean of two positive numbers is the positive
square root of their product. So the geometric mean of a and b is
the positive number x such
that
Holt Geometry
, or x2 = ab.
10. 8-1 Similarity in Right Triangles
Example 2A: Finding Geometric Means
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
4 and 25
Let x be the geometric mean.
x2 = (4)(25) = 100
x = 10
Holt Geometry
Def. of geometric mean
Find the positive square root.
11. 8-1 Similarity in Right Triangles
Example 2B: Finding Geometric Means
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
5 and 30
Let x be the geometric mean.
x2 = (5)(30) = 150
Def. of geometric mean
Find the positive square root.
Holt Geometry
12. 8-1 Similarity in Right Triangles
Check It Out! Example 2a
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
2 and 8
Let x be the geometric mean.
x2 = (2)(8) = 16
x=4
Holt Geometry
Def. of geometric mean
Find the positive square root.
13. 8-1 Similarity in Right Triangles
Check It Out! Example 2b
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
10 and 30
Let x be the geometric mean.
x2 = (10)(30) = 300
Def. of geometric mean
Find the positive square root.
Holt Geometry
14. 8-1 Similarity in Right Triangles
Check It Out! Example 2c
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
8 and 9
Let x be the geometric mean.
x2 = (8)(9) = 72
Def. of geometric mean
Find the positive square root.
Holt Geometry
15. 8-1 Similarity in Right Triangles
You can use Theorem 8-1-1 to write proportions comparing
the side lengths of the triangles formed by the altitude to the
hypotenuse of a right triangle.
All the relationships in red involve geometric means.
Holt Geometry
17. 8-1 Similarity in Right Triangles
Example 3: Finding Side Lengths in Right Triangles
Find x, y, and z.
62 = (9)(x)
x=4
y2 = (4)(13) = 52
6 is the geometric mean of 9
and x.
Divide both sides by 9.
y is the geometric mean of 4
and 13.
Find the positive square root.
z2 = (9)(13) = 117
Holt Geometry
z is the geometric mean of
9 and 13.
Find the positive square root.
18. 8-1 Similarity in Right Triangles
Helpful Hint
Once you’ve found the unknown side lengths, you can
use the Pythagorean Theorem to check your answers.
Holt Geometry
19. 8-1 Similarity in Right Triangles
Check It Out! Example 3
Find u, v, and w.
92 = (3)(u)
9 is the geometric mean of
u and 3.
u = 27
Divide both sides by 3.
w2 = (27 + 3)(27) w is the geometric mean of
u + 3 and 27.
Find the positive square root.
v2 = (27 + 3)(3)
Holt Geometry
v is the geometric mean of
u + 3 and 3.
Find the positive square root.
20. 8-1 Similarity in Right Triangles
Example 4: Measurement Application
To estimate the height of a
Douglas fir, Jan positions herself
so that her lines of sight to the
top and bottom of the tree form a
90º angle. Her eyes are about 1.6
m above the ground, and she is
standing 7.8 m from the tree.
What is the height of the tree to
the nearest meter?
Holt Geometry
21. 8-1 Similarity in Right Triangles
Example 4 Continued
Let x be the height of the tree above eye level.
(7.8)2 = 1.6x
x = 38.025 ≈ 38
7.8 is the geometric mean of
1.6 and x.
Solve for x and round.
The tree is about 38 + 1.6 = 39.6, or 40 m tall.
Holt Geometry
22. 8-1 Similarity in Right Triangles
Check It Out! Example 4
A surveyor positions himself so
that his line of sight to the top of a
cliff and his line of sight to the
bottom form a right angle as
shown.
What is the height of the cliff to
the nearest foot?
Holt Geometry
23. 8-1 Similarity in Right Triangles
Check It Out! Example 4 Continued
Let x be the height of cliff above eye level.
(28)2 = 5.5x
x
142.5
28 is the geometric mean of
5.5 and x.
Divide both sides by 5.5.
The cliff is about 142.5 + 5.5, or 148 ft
high.
Holt Geometry
24. 8-1 Similarity in Right Triangles
Lesson Quiz: Part I
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
1. 8 and 18
2. 6 and 15
Holt Geometry
12
25. 8-1 Similarity in Right Triangles
Lesson Quiz: Part II
For Items 3–6, use ∆RST.
3. Write a similarity statement comparing the three
triangles.
∆RST ~ ∆RPS ~ ∆SPT
4. If PS = 6 and PT = 9, find PR.
4
5. If TP = 24 and PR = 6, find RS.
6. Complete the equation (ST)2 = (TP + PR)(?).
Holt Geometry
TP