Prove and apply properties of rectangles, rhombuses, and squares.
Use properties of rectangles, rhombuses, and squares to solve problems.
Prove that a given quadrilateral is a rectangle, rhombus, or square.
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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.
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the SquareJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
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.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
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Model Attribute Check Company Auto PropertyCeline George
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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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
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
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!
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
1. Obj. 26 Special Parallelograms
The student is able to (I can):
• Prove and apply properties of rectangles, rhombuses, and
squares.
• Use properties of rectangles, rhombuses, and squares to
solve problems.
• Prove that a given quadrilateral is a rectangle, rhombus,
or square.
2. rectangle
A parallelogram with four right angles.
If a parallelogram is a rectangle, then its
diagonals are congruent (“checking for
square”).
F
I
FS ≅ IH
H
S
3. Because a rectangle is a parallelogram, it
also “inherits” all of the properties of a
parallelogram:
• Opposite sides parallel
• Opposite sides congruent
• Opposite angles congruent (actually all
angles are congruent)
• Consecutive angles supplementary
• Diagonals bisect each other
4. Example
Find each length.
1. LW
LW = FO = 30
F
30
O
17
L
2. OL
OL = FW = 2(17) = 34
3. OW
∆OWL is a right triangle, so
OW 2 + LW 2 = OL2
OW 2 + 302 = 34 2
OW 2 + 900 = 1156
OW 2 = 256
OW = 16
W
5. rhombus
A parallelogram with four congruent sides.
If a parallelogram is a rhombus, then its
diagonals are perpendicular.
6. Proof:
B
O
S
L
W
Because BOWL is a rhombus, BO ≅ OW.
Diagonals bisect each other, so BS ≅ WS.
The reflexive property means that OS ≅ OS.
Therefore, ∆OSB ≅ ∆OSW by SSS. This
means that ∠OSB ≅ ∠OSW. Since they
are also supplementary, they must be 90º.
7. If a parallelogram is a rhombus, then each
diagonal bisects a pair of opposite angles.
3
1 2
8
∠1 ≅ ∠2
∠3 ≅ ∠4
∠5 ≅ ∠6
∠7 ≅ ∠8
7
4
6
5
Since opposite angles are
also congruent:
∠1 ≅ ∠2 ≅ ∠5 ≅ ∠6
∠3 ≅ ∠4 ≅ ∠7 ≅ ∠8
8. Examples
1. What is the perimeter of a rhombus
whose side length is 7?
4(7) = 28
2. Find the value of x
The side = 10
Pyth. triple: 6, 8, 10
x=6
Perimeter = 40
(13y—9)º
3. Find the value of y
13y — 9 = 3y + 11
10y = 20
y=2
x
10
8
(3y+11)º
9. square
A quadrilateral with four right angles and
four congruent sides.
Note: A square has all of the properties of
both a rectangle and a rhombus:
• Diagonals are congruent
• Diagonals are perpendicular
• Diagonals bisect opposite angles.
10. Conditions for
Special
Parallelograms
You can always use the definitions to
prove these, but there are also some
shortcuts we can use. For all of these
shortcuts, we must first prove or know
that the quadrilateral is a parallelogram.
• To prove a parallelogram is a rectangle
(pick one):
— One angle is a right angle
— The diagonals are congruent
11. • To prove a parallelogram is a rhombus
(pick one):
— A pair of consecutive sides is
congruent
— The diagonals are perpendicular
— One diagonal bisects a pair of
opposite angles
• To prove that a quadrilateral is a
square:
— It is both a rectangle and a rhombus.
