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An Introduction to Geometry

In geometry, a geometric object is a set of points, lines, or polygons, and the relations between them, in any notion of location or configuration, such as a plane figure, space, or a higher-dimensional space of an extended sort. Geometry has many subfields, including projective geometry, descriptive geometry, algebraic geometry, and combinatorial geometry. Geometry is one of the oldest fields of knowledge, and is included in the broad field of mathematics. The most basic questions in geometry are where, when, and why these geometries exist. Geometry can be divided into two main branches: Euclidean geometry and non-Euclidean geometry. Euclidean geometry is the branch of mathematics dealing with conic sections, and the geometry of the Euclidean plane and space, while non-Euclidean geometry is the branch of mathematics concerning the properties of these kinds of geometries. The field of geometry is one of the oldest fields of study. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia in the first millennium BC. The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, which date from the Third Dynasty of Egypt and the Second Dynasty of Ancient Egypt, respectively. Both texts on Egyptian geometry made use of geometry in their demonstration of many of their geometrical propositions. They were used by the Egyptians during the Middle Kingdom (2000–1800 BC) and later on in New Kingdom Egypt (1550–1070 BC) and were also used in the Second Intermediate Period (1070–945 BC). The ancient Greeks started to formalize geometry during the 5th century BC. They used geometry to calculate the lengths of the sides of a polygon, and to draw its interior. They used geometry to calculate the circumference of a circle, and to draw its circumference and its diameter. They used geometry to calculate the area and the volume of a figure, and to calculate the surface area and the volume of a solid. In ancient Greek philosophy, geometry was used to define nature and the physical universe, and also used to describe the human mind. Euclid's "Elements" helped to establish geometry as a separate branch of science. The Indian mathematician Mahavira (c. 6th century BC) was the first to state explicitly the principles of what is now called Euclidean geometry. He described the 5th century BC plane geometry of Eudoxus of Cnidus in a detailed commentary on the 13th century BC Indian philosophical work "Yukti-sastra" of the ancient Indian mathematicians Katyayana and Panini. The 9th century mathematician Brahmagupta used geometry to calculate the volume of a triangle and the area of a circle. He provided the first correct proofs for many geometric theorems. In the Islamic world, the first person to work on geometry fully was the Islamic polymath al-Haytham (Alhazen) (965–1040 AD). He was strongly influenced by Greek philosophy and in turn influenced the further development of geometry in Europe. In his book "Al-manasir wa al-hal", he discussed the general principles of proportion. In the book "Flatness of the Earth and its Rectification" he investigated the sphericity of the Earth and its possible rectification. In his book "The Configuration of the Universe" he proposed a model of the solar system, which was later used and developed by Ptolemy. The book also contains the first good description of the segment of a parabola and of a cycloid that can be found in any modern text book on geometry. Al-Haytham's book "Commentary on the Posterior" was a commentary on Ptolemy's "Geography" and was the first Arabic book to include a section on mathematics. In the 11th century, the Persian scholar Abu Nasr Mansur ibn al-Farkadain (also known as Albatenius) wrote a book called "Matheseos libri septem", which was the first book on mathematics written in Arabic. Abu Nasr's book included many mathematical results, including the first proof of the Pythagorean theorem. In the 13th century, the Italian mathematician Fibonacci (c. 1170–1250 AD) wrote his "Liber Abaci" ("Book of the Abacus"), a widely used mathematics book, which became a standard arithmetic text in Europe. In 1225, the Pisan mathematician Leonardo of Pisa (Leonardo Fibonacci) (Leonardo da Pisa) published his "Liber Quadratorum" ("Book of the Four Square") in Bologna. In 1238, the French philosopher and mathematician Nicole Oresme (c. 13

Basic Geometric Figures; Angles

176 Practice Problems
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01:09
Precalculus

Find an equation of the line that passes through the point (2,6) in such a way that the segment of the line cut off between the axes is bisected by the point (2,6).

The Conic Sections
The Basic Equations
Sriram Soundarrajan
02:33
Precalculus

From the point $(7,-1),$ tangent lines are drawn to the circle $(x-4)^{2}+(y-3)^{2}=4 .$ Find the slopes of these lines.

The Conic Sections
The Basic Equations
Sriram Soundarrajan
00:58
Precalculus

Find the distance from the point to the line using: (a) the formula $d=\left|m x_{0}+b-y_{0}\right| / \sqrt{1+m^{2}} ;$ and (b) the formula $d=\left|A x_{0}+B y_{0}+C\right| / \sqrt{A^{2}+B^{2}}$.
$$(1,4); y=x-2$$

The Conic Sections
The Basic Equations
Sriram Soundarrajan

Parallel and Perpendicular Lines

105 Practice Problems
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02:52
Understanding Elementary Algebra with Geometry

In the following figure, $L_{1}$ is parallel to $L_{2}, L_{3}$ is parallel to $L_{4},$ and $\angle 2=40^{\circ} .$ Find all the remaining angles. (FIGURE CANNOT COPY)

Geometry
Angles
Julie Silva
01:24
Understanding Elementary Algebra with Geometry

In the following figure $\angle 4=25^{\circ} .$ Find the measures of angles $1,2,$ and $3 .$ (FIGURE CANNOT COPY)

Geometry
Angles
Julie Silva
03:43
Geometry for Enjoyment and Challenge

Problem Set B, continued
Given: $\overline{\mathrm{AB}} \perp \mathrm{m}$equilateral $\triangle$ DBC lies in plane m.
Prove: $\triangle \mathrm{ACD}$ is isosceles.
FIGURE CAN'T COPY)

Lines and Planes in Space
Perpendicularity of a Line and a Plane
Debasish Das

Triangles

198 Practice Problems
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06:38
Precalculus

Law of Tangents For any triangle, derive the Law of Tangents.
$$
\frac{a-b}{a+b}=\frac{\tan \left[\frac{1}{2}(A-B)\right]}{\tan \left[\frac{1}{2}(A+B)\right]}
$$
[Hint: Use Mollweide's Formula.]

Applications of Trigonometric Functions
The Law of sines
Mitchell Cutler
02:28
Precalculus

An awning that covers a sliding glass door that is 88 inches tall forms an angle of $50^{\circ}$ with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the Sun is more than $65^{\circ} .$ See the figure. Find the length $L$ of the awning.
(FIGURE CANNOT COPY)

Applications of Trigonometric Functions
The Law of sines
Mitchell Cutler
03:53
Precalculus

Solve each triangle using either the Law of Sines or the Law of Cosines.
$$b=5, c=12, A=60^{\circ}$$

Applications of Trigonometric Functions
The Law of Cosines
Lourence Gonhovi

The Pythagorean Theorem

68 Practice Problems
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04:59
Precalculus

Use the diagram given to derive a formula for the height $h$ of the taller building in terms of the height $x$ of the shorter building and the ratios
for tan $u$ and tan $v .$ Then use the formula to find $h$ given the shorter building is $75 \mathrm{m}$ tall with $u=40^{\circ}$ and $v=50^{\circ}$
GRAPH CANT COPY

An Introduction to Trigonometric Functions
The Trigonometry of Right Triangles
Tani Iqbal
03:49
Precalculus

Angle of elevation: The tallest free-standing tower in the world is the CNN Tower in Toronto, Canada. The tower includes a rotating restaurant high above the ground. From a distance of $500 \mathrm{ft}$ the angle of elevation to the pinnacle of the tower is $74.6^{\circ} .$ The angle of elevation to the restaurant from the same vantage point is $66.5^{\circ} .$ How tall is the CNN Tower? How far below the pinnacle of the tower is the restaurant located?
GRAPH CANT COPY

An Introduction to Trigonometric Functions
The Trigonometry of Right Triangles
Katelyn Chen
01:51
Precalculus

Angle of depression: A person standing near the top of the Eiffel Tower notices a car wreck some distance from the tower. If the angle of depression from the person's eyes to the wreck is $32^{\circ},$ how far away is the accident from the base of the tower? See Exercise 71

An Introduction to Trigonometric Functions
The Trigonometry of Right Triangles
Katelyn Chen

Congruent Triangles and Similar Triangles

40 Practice Problems
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03:06
Understanding Elementary Algebra with Geometry

In Exercises $27-34,$ find the length of the missing sides of the given right triangles.
(TRIANGLE CAN'T COPY).

Geometry
Similarity
Carrie Bain
01:58
Understanding Elementary Algebra with Geometry

In Exercises $21-26,$ round your answer to the nearest tenth where necessary.
The corresponding sides of two similar triangles are in the ratio of 4 to $7 .$ If a side of the smaller triangle is $5.8 \mathrm{cm},$ find the length of the corresponding side of the larger triangle.

Geometry
Similarity
Carrie Bain
02:11
Understanding Elementary Algebra with Geometry

In Exercises $11-14,$ find the length of the indicated side.
Find $|\overline{A C}|$
(IMAGE CAN'T COPY).

Geometry
Similarity
Carrie Bain

Quadrilaterals and Other Polygons

62 Practice Problems
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01:52
Understanding Elementary Algebra with Geometry

Find the length of the sides of a rhombus with diagonals $12^{\prime \prime}$ and $18^{\prime \prime}$.

Geometry
Quadrilaterals
Sriram Soundarrajan
00:53
Understanding Elementary Algebra with Geometry

Find the missing angles of the quadrilateral.

Geometry
Quadrilaterals
Sriram Soundarrajan
02:10
Geometry for Enjoyment and Challenge

Answer Always, Sometimes, or Never: A quadrilateral is a parallelogram if
a Diagonals are congruent
b One pair of opposite sides are congruent and one pair of opposite sides are parallel
c Each pair of consecutive angles are supplementary
d All angles are right angles

Parallel Lines and Related Figures
Proving That a Quadrilateral Is a Parallelogram
Aparna Shakti

Perimeters and Areas of Polygons

53 Practice Problems
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01:44
Understanding Elementary Algebra with Geometry

A side of one polygon is $15 \mathrm{ft}$ and the corresponding side of a second similar polygon is 2 ft. If the perimeter of the first polygon is 80 ft and its area is 2000 sq fi. find the perimeter and area of the similar polygon.

Geometry
Perimeter and Area
Sriram Soundarrajan
01:44
Understanding Elementary Algebra with Geometry

A side of one triangle is 28 in. and the corresponding side of a second similar triangle is 42 in... If the perimeter of the first triangle is 98 in. and its area is
$420 \mathrm{sq}$ in.., find the perimeter and area of the similar triangle.

Geometry
Perimeter and Area
Sriram Soundarrajan
01:28
Understanding Elementary Algebra with Geometry

Find the perimeter and area of the given figure.

Geometry
Perimeter and Area
Sriram Soundarrajan

Circles

205 Practice Problems
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02:00
Precalculus

Find the angle, radius, are length, and/or area as needed, until all values are known. (FIGURE CAN'T COPY)

An Introduction to Trigonometric Functions
Angle Measure, Special Triangles, and Special Angles
Lourence Gonhovi
01:31
Precalculus

Find the angle, radius, are length, and/or area as needed, until all values are known. (FIGURE CAN'T COPY)

An Introduction to Trigonometric Functions
Angle Measure, Special Triangles, and Special Angles
Lourence Gonhovi
01:24
Precalculus

Use the formula for area of a circular sector to find the value of the unknown quantity: $A=\frac{1}{2} r^{2} \theta$.
$$\theta=5 ; r=6.8 \mathrm{km}$$

An Introduction to Trigonometric Functions
Angle Measure, Special Triangles, and Special Angles
Lourence Gonhovi

Volume

305 Practice Problems
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01:52
Physical Chemistry

The densities of liquid and vapor methyl ether in $\mathrm{g} \mathrm{cm}^{-3}$ at various temperatures are as follows:
$$\begin{array}{cccccc}
t /^{\circ} \mathrm{C} & 30 & 50 & 70 & 100 & 120 \\
\rho_{1} & 0.6455 & 0.6116 & 0.5735 & 0.4950 & 0.4040 \\
\rho_{\mathrm{v}} & 0.0142 & 0.0241 & 0.0385 & 0.0810 & 0.1465\end{array}$$
The critical temperature of methyl ether is $299^{\circ} \mathrm{C}$. What is the critical molar volume? (See Problem $1.8 .)$

Zeroth Law of Thermodynamics and Equations of State
Crystal Wang
01:54
Engineering Mechanics: Statics and Dynamics

The sphere is formed by revolving the shaded area around the $x$ axis. Determine the moment of inertia $I_{x}$ and express the result in terms of the total mass $m$ of the sphere. The material has a constant density $\rho$.

Planar Kinetics of a Rigid Body: Force and Acceleration
Ahmed Kamel
01:01
Engineering Mechanics: Statics and Dynamics

Determine the moment of inertia $I_{y}$ for the slender rod. The rod's density $\rho$ and cross-sectional area $A$ are constant. Express the result in terms of the rod's total mass $m$.

Planar Kinetics of a Rigid Body: Force and Acceleration
Ahmed Kamel

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