# 13.4: Math Models and Geometry

We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin by considering a formal approach to solving problems and use it to solve a variety of common problems, including making decisions about money. Then we will explore geometry and relate it to everyday situations, using the problem-solving strategy we develop.

• 13.4.1: Solve Money Applications
Solving coin word problems is much like solving any other word problem. However, what makes them unique is that you have to find the total value of the coins instead of just the total number of coins. For coins of the same type, the total value can be found by multiplying the number of coins by the value of an individual coin. You may find it helpful to put all the numbers into a table to make sure they check.
• 13.4.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 1)
An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. If the sum of the measures of two angles is 180°, then they are supplementary angles. But if their sum is 90°, then they are complementary angles. We will adapt our Problem Solving Strategy for Geometry Applications. Since these applications will involve geometric shapes, it will help to draw a figure and label it with the information from the problem.
• 13.4.3: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)
Triangles are named by their vertices. For any triangle, the sum of the measures of the angles is 180°. Some triangles have special names such as the right triangle which has one 90° angle. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. To solve problems that use the Pythagorean Theorem, we will need to find square roots.
• 13.4.4: Use Properties of Rectangles, Triangles, and Trapezoids (Part 1)
Many geometry applications will involve finding the perimeter or the area of a figure. The perimeter is a measure of the distance around a figure. The area is a measure of the surface covered by a figure. The volume is a measure of the amount of space occupied by a figure. A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and the adjacent side as the width, W.
• 13.4.5: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2)
Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of a triangle is one-half the base times the height. An isosceles triangle is a triangle with two sides of equal length is while a triangle that has three sides of equal length is an equilateral triangle. A trapezoid is four-sided figure with two sides that are parallel, the bases, and two sides that are not. The area of a trapezoid is one-half the height times the sum of the bases.

Figure 9.1 - Note the many individual shapes in this building. (credit: Bert Kaufmann, Flickr)

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### Based on the TExES 4-8 Mathematics Standards

The University of Houston's free online mathematics quizzes are based on the TExES Mathematics 4-8 Domains and Competencies, which are listed below. For information about how each online quiz relates to these competencies, click on the "Overview of Quizzes" link in the navigation pane at the left.

THE TEACHER UNDERSTANDS THE STRUCTURE OF NUMBER SYSTEMS, THE DEVELOPMENT OF A SENSE OF QUANTITY AND THE RELATIONSHIP BETWEEN QUANTITY AND SYMBOLIC REPRESENTATIONS.

1. Analyzes the structure of numeration systems and the roles of place value and zero in the base ten system.

THE TEACHER UNDERSTANDS NUMBER OPERATIONS AND COMPUTATIONAL ALGORITHMS.

1. Works proficiently with real and complex numbers and their operations.

THE TEACHER UNDERSTANDS IDEAS OF NUMBER THEORY AND USES NUMBERS TO MODEL AND SOLVE PROBLEMS WITHIN AND OUTSIDE OF MATHEMATICS.

1. Demonstrates an understanding of ideas from number theory (e.g., prime factorization, greatest common divisor) as they apply to whole numbers, integers and rational numbers and uses these ideas in problem situations.

DOMAIN II — PATTERNS AND ALGEBRA

THE TEACHER UNDERSTANDS AND USES MATHEMATICAL REASONING TO IDENTIFY, EXTEND AND ANALYZE PATTERNS AND UNDERSTANDS THE RELATIONSHIPS AMONG VARIABLES, EXPRESSIONS, EQUATIONS, INEQUALITIES, RELATIONS AND FUNCTIONS.

1. Uses inductive reasoning to identify, extend and create patterns using concrete models, figures, numbers and algebraic expressions.

THE TEACHER UNDERSTANDS AND USES LINEAR FUNCTIONS TO MODEL AND SOLVE PROBLEMS.

1. Demonstrates an understanding of the concept of linear function using concrete models, tables, graphs and symbolic and verbal representations.

THE TEACHER UNDERSTANDS AND USES NONLINEAR FUNCTIONS AND RELATIONS TO MODEL AND SOLVE PROBLEMS.

1. Uses a variety of methods to investigate the roots (real and complex), vertex and symmetry of a quadratic function or relation.

THE TEACHER USES AND UNDERSTANDS THE CONCEPTUAL FOUNDATIONS OF CALCULUS RELATED TO TOPICS IN MIDDLE SCHOOL MATHEMATICS.

1. Relates topics in middle school mathematics to the concept of limit in sequences and series.

THE TEACHER UNDERSTANDS MEASUREMENT AS A PROCESS.

1. Selects and uses appropriate units of measurement (e.g., temperature, money, mass, weight, area, capacity, density, percents, speed, acceleration) to quantify, compare and communicate information.

THE TEACHER UNDERSTANDS THE GEOMETRIC RELATIONSHIPS AND AXIOMATIC STRUCTURE OF EUCLIDEAN GEOMETRY.

1. Understands concepts and properties of points, lines, planes, angles, lengths and distances.

THE TEACHER ANALYZES THE PROPERTIES OF TWO- AND THREE-DIMENSIONAL FIGURES.

1. Uses and understands the development of formulas to find lengths, perimeters, areas and volumes of basic geometric figures.

THE TEACHER UNDERSTANDS TRANSFORMATIONAL GEOMETRY AND RELATES ALGEBRA TO GEOMETRY AND TRIGONOMETRY USING THE CARTESIAN COORDINATE SYSTEM.

1. Describes and justifies geometric constructions made using a reflection device and other appropriate technologies.

THE TEACHER UNDERSTANDS HOW TO USE GRAPHICAL AND NUMERICAL TECHNIQUES TO EXPLORE DATA, CHARACTERIZE PATTERNS AND DESCRIBE DEPARTURES FROM PATTERNS.

1. Organizes and displays data in a variety of formats (e.g., tables, frequency distributions, stem-and-leaf plots, box-and-whisker plots, histograms, pie charts).

THE TEACHER UNDERSTANDS THE THEORY OF PROBABILITY.

1. Explores concepts of probability through data collection, experiments and simulations.

THE TEACHER UNDERSTANDS THE RELATIONSHIP AMONG PROBABILITY THEORY, SAMPLING AND STATISTICAL INFERENCE AND HOW STATISTICAL INFERENCE IS USED IN MAKING AND EVALUATING PREDICTIONS.

1. Applies knowledge of designing, conducting, analyzing and interpreting statistical experiments to investigate real-world problems.

THE TEACHER UNDERSTANDS MATHEMATICAL REASONING AND PROBLEM SOLVING.

1. Demonstrates an understanding of proof, including indirect proof, in mathematics.

THE TEACHER UNDERSTANDS MATHEMATICAL CONNECTIONS WITHIN AND OUTSIDE OF MATHEMATICS AND HOW TO COMMUNICATE MATHEMATICAL IDEAS AND CONCEPTS.

1. Recognizes and uses multiple representations of a mathematical concept (e.g., a point and its coordinates, the area of circle as a quadratic function in r, probability as the ratio of two areas).

THE TEACHER UNDERSTANDS HOW CHILDREN LEARN AND DEVELOP MATHEMATICAL SKILLS, PROCEDURES AND CONCEPTS.

1. Applies theories and principles of learning mathematics to plan appropriate instructional activities for all students.

THE TEACHER UNDERSTANDS HOW TO PLAN, ORGANIZE AND IMPLEMENT INSTRUCTION USING KNOWLEDGE OF STUDENTS, SUBJECT MATTER AND STATEWIDE CURRICULUM (TEXAS ESSENTIAL KNOWLEDGE AND SKILLS [TEKS]) TO TEACH ALL STUDENTS TO USE MATHEMATICS.

1. Demonstrates an understanding of a variety of instructional methods, tools and tasks that promote students’ ability to do mathematics described in the TEKS.

THE TEACHER UNDERSTANDS ASSESSMENT AND USES A VARIETY OF FORMAL AND INFORMAL ASSESSMENT TECHNIQUES TO MONITOR AND GUIDE MATHEMATICS INSTRUCTION AND TO EVALUATE STUDENT PROGRESS.

1. Demonstrates an understanding of the purpose, characteristics and uses of various assessments in mathematics, including formative and summative assessments.

## 13.4: Math Models and Geometry

Most indoor mobile robots do not move like a car. For example, consider the mobile robotics platform shown in Figure 13.2a. This is an example of the most popular way to drive indoor mobile robots. There are two main wheels, each of which is attached to its own motor. A third wheel (not visible in Figure 13.2a) is placed in the rear to passively roll along while preventing the robot from falling over.

Figure 13.2: (a) The Pioneer 3-DX8 (courtesy of ActivMedia Robotics: MobileRobots.com), and many other mobile robots use a differential drive. In addition to the two drive wheels, a caster wheel (as on the bottom of an office chair) is placed in the rear center to prevent the robot from toppling over. (b) The parameters of a generic differential-drive robot.
Figure 13.3: (a) Pure translation occurs when both wheels move at the same angular velocity (b) pure rotation occurs when the wheels move at opposite velocities.

To construct a simple model of the constraints that arise from the differential drive, only the distance between the two wheels, and the wheel radius, , are necessary. See Figure 13.2b. The action vector directly specifies the two angular wheel velocities (e.g., in radians per second). Consider how the robot moves as different actions are applied. See Figure 13.3. If 0\$ --> , then the robot moves forward in the direction that the wheels are pointing. The speed is proportional to . In general, if , then the distance traveled over a duration of time is (because is the total angular displacement of the wheels). If , then the robot rotates clockwise because the wheels are turning in opposite directions. This motivates the placement of the body-frame origin at the center of the axle between the wheels. By this assignment, no translation occurs if the wheels rotate at the same rate but in opposite directions.

Based on these observations, the configuration transition equation is

The translational part contains and parts, just like the simple car because the differential drive moves in the direction that its drive wheels are pointing. The translation speed depends on the average of the angular wheel velocities. To see this, consider the case in which one wheel is fixed and the other rotates. This initially causes the robot to translate at of the speed in comparison to both wheels rotating. The rotational speed is proportional to the change in angular wheel speeds. The robot's rotation rate grows linearly with the wheel radius but reduces linearly with respect to the distance between the wheels.

It is sometimes preferable to transform the action space. Let and . In this case, can be interpreted as an action variable that means translate,'' and means rotate.'' Using these actions, the configuration transition equation becomes

In this form, the configuration transition equation resembles (13.15) for the simple car (try setting and ). A differential drive can easily simulate the motions of the simple car. For the differential drive, the rotation rate can be set independently of the translational velocity. The simple car, however, has the speed appearing in the expression. Therefore, the rotation rate depends on the translational velocity.

Figure 13.4: The shortest path traversed by the center of the axle is simply the line segment that connects the initial and goal positions in the plane. Rotations appear to be cost-free.

Recall the question asked about shortest paths for the Reeds-Shepp and Dubins cars. The same question for the differential drive turns out to be uninteresting because the differential drive can cause the center of its axle to follow any continuous path in . As depicted in Figure 13.4, it can move between any two configurations by: 1) first rotating itself to point the wheels to the goal position, which causes no translation 2) translating itself to the goal position and 3) rotating itself to the desired orientation, which again causes no translation. The total distance traveled by the center of the axle is always the Euclidean distance in between the two desired positions.

This may seem like a strange effect due to the placement of the coordinate origin. Rotations seem to have no cost. This can be fixed by optimizing the total amount of wheel rotation or time required, if the speed is held fixed . Suppose that . Determining the minimum time required to travel between two configurations is quite interesting and is covered in Section 15.3. This properly takes into account the cost of rotating the robot, even if it does not cause a translation.

## Select Grade 8 Math Worksheets by Topic

Explore 2,400+ Eighth Grade Math Worksheets Convert each fraction with a multiple of 10 as its denominator into a decimal number by placing the decimal point at the right spot. Apply prime factorization and determine the square roots of the first fifty perfect squares offered as positive integers. Use the formula, m = (y2 - y1) / (x1 - x1) to find the slope(m) of a line passing through two points: (x1,y1) and (x2,y2). Follow the order of operations, rearrange to make the unknown variable the subject, and solve for its integer value. Observe each set of ordered pairs given in Part A, figure out ordered pairs from graphs in Part B, and state if they represent a function. Slide each figure in the said direction: up or down, left or right. Write the coordinates of the shifted image. Complete the congruence statement for each pair of triangles by writing the corresponding side or corresponding angle. Find the measure of the indicated interior angle by subtracting the sum of the known angles from 180. Observe whether the interior angles lie on the same side or opposite sides of the transversal and find the unknown angle. Square the adjacent and opposite sides of the triangle take the root of their sum if you arrive at the hypotenuse, then it's a right triangle. Plug the given radius(r) and height(h) in the formula V = 1/3πr 2 h and find the volume of the cone. Read each word problem with a real-life scenario and find the mean, median, mode, and range for each data set. Switch each fraction to percent by multiplying the numerator by 100, dividing the product by the denominator, and adding the % symbol. The square of a square root is the radicand. So, simply multiply the radicand with the square of the number outside the root. Isolate the x and y-terms to one side and the constant to the other side of the equation and rewrite it in the form: ax + by = c.

## Contents

### Ancient Greece Edit

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry. 

Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension with the question of finding points on a line that were in a ratio to the others.  Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation. 

### Persia Edit

The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra  with his geometric solution of the general cubic equations,  but the decisive step came later with Descartes.  Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe.  Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.  : 248

### Western Europe Edit

Analytic geometry was independently invented by René Descartes and Pierre de Fermat,   although Descartes is sometimes given sole credit.   Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. La Geometrie, written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition. 

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse.    Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.  As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following: 

### Cartesian coordinates (in a plane or space) Edit

The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).

### Polar coordinates (in a plane) Edit

In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ, with θ normally measured counterclockwise from the positive x-axis. Using this notation, points are typically written as an ordered pair (r, θ). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: x = r cos ⁡ θ , y = r sin ⁡ θ r = x 2 + y 2 , θ = arctan ⁡ ( y / x ) +y^<2>>>,, heta =arctan(y/x)> . This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates.

### Cylindrical coordinates (in a space) Edit

In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis.

### Spherical coordinates (in a space) Edit

In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics. 

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures. 

Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x 2 + y 2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations.  The equation x 2 + y 2 = r 2 is the equation for any circle centered at the origin (0, 0) with a radius of r.

### Lines and planes Edit

Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

m is the slope or gradient of the line. b is the y-intercept of the line. x is the independent variable of the function y = f(x).

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

(The dot here means a dot product, not scalar multiplication.) Expanded this becomes

which is the point-normal form of the equation of a plane.  This is just a linear equation:

Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation

is a plane having the vector n = ( a , b , c ) =(a,b,c)> as a normal.  This familiar equation for a plane is called the general form of the equation of the plane. 

In three dimensions, lines can not be described by a single linear equation, so they are frequently described by parametric equations:

x, y, and z are all functions of the independent variable t which ranges over the real numbers. (x0, y0, z0) is any point on the line. a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line.

### Conic sections Edit

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form

As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space P 5 . ^<5>.>

The conic sections described by this equation can be classified using the discriminant 

If the conic is non-degenerate, then:

• if B 2 − 4 A C < 0 -4AC<0> , the equation represents an ellipse
• if A = C and B = 0 , the equation represents a circle, which is a special case of an ellipse
• if we also have A + C = 0 , the equation represents a rectangular hyperbola.

A quadric, or quadric surface, is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x1, x2,x3 , the general quadric is defined by the algebraic equation 

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula

which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

where m is the slope of the line.

In three dimensions, distance is given by the generalization of the Pythagorean theorem:

while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by 

where θ is the angle between A and B.

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.

Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.

is the relation that describes the unit circle.

Traditional methods for finding intersections include substitution and elimination.

So our intersection has two points:

So our intersection has two points:

For conic sections, as many as 4 points might be in the intersection.

### Finding intercepts Edit

One type of intersection which is widely studied is the intersection of a geometric object with the x and y coordinate axes.

### Tangent lines and planes Edit

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f ' (c) where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized see Tangent space.

### Normal line and vector Edit

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

## What is the Thales Theorem?

The Thales theorem states that:

If three points A, B, and C lie on the circumference of a circle, whereby the line AC is the diameter of the circle, then the angle ABC is a right angle (90°).

Alternatively, we can state the Thales theorem as:

The diameter of a circle always subtends a right angle to any point on the circle. You noticed that the Thales theorem is a special case of the inscribed angle theorem (the central angle = twice the inscribed angle).

Thales theorem is attributed to Thales, a Greek mathematician and philosopher who was based in Miletus. Thales first initiated and formulated the Theoretical Study of Geometry to make astronomy a more exact science.

There are multiple ways to prove Thales Theorem. We can use geometry and algebra techniques to prove this theorem. Since this is a geometry topic, therefore, let’s see the most basic method below.

## New Jersey Department of Education Moving towards formal mathematical arguments, the standards presented in this high school geometry course are meant to formalize and extend middle grades geometric experiences. Transformations are presented early in the year to assist with the building of conceptual understandings of the geometric concepts.

In unit 1, triangle congruence conditions are established using analysis of rigid motion and formal constructions. Various formats will be used to prove theorems about angles, lines, triangles and other polygons. The work in unit 2 will build on the students understanding of dilations and proportional reasoning to develop a formal understanding of similarity.

The standards included in unit 3 extend the notion of similarity to right triangles and the understanding of right triangle trigonometry. In developing the Laws of Sines and Cosines, the students are expected to find missing measures of triangles in general, not just right triangles.

Work in unit 4 will focus on circles and using the rectangular coordinate system to verify geometric properties and to solve geometric problems. Concepts of similarity will be used to establish the relationship among segments on chords, secants and tangents as well as to prove basic theorems about circles.

The standards in unit 5 will extend previous understandings of two- dimensional objects in order to explain, visualize, and apply geometric concepts to three-dimensional objects. Informal explanations of circumference, area and volume formulas will be analyzed.