Triangles

12 Key excerpts on "Triangles"

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  eBook - ePub

  Geometry For Dummies

  • Mark Ryan(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Part 3

  Triangles: Polygons of the Three-Sided Variety

  IN THIS PART … Get familiar with triangle basics. Have fun with right Triangles. Work on congruent triangle proofs. Passage contains an image Chapter 7

  Grasping Triangle Fundamentals

  IN THIS CHAPTER Looking at a triangle’s sides: Equal or unequal Uncovering the triangle inequality principle Classifying Triangles by their angles Calculating the area of a triangle Finding the four “centers” of a triangle

  Considering that it’s the runt of the polygon family, the triangle sure does play a big role in geometry. Triangles are one of the most important components of geometry proofs (you see triangle proofs in Chapter 9 ). They also have a great number of interesting properties that you might not expect from the simplest possible polygon. Maybe Leonardo da Vinci (1452–1519) was on to something when he said, “Simplicity is the ultimate sophistication.”

  In this chapter, I take you through the triangle basics — their names, sides, angles, and area. I also show you how to find the four “centers” of a triangle.

  Taking In a Triangle’s Sides

  Triangles are classified according to the length of their sides or the measure of their angles. These classifications come in threes, just like the sides and angles themselves. That is, a triangle has three sides, and three terms describe Triangles based on their sides; a triangle also has three angles, and three classifications of Triangles are based on their angles. I talk about classifications based on angles in the upcoming section “Getting to Know Triangles by Their Angles .”

  The following are triangle classifications based on sides:

  • Scalene triangle: A scalene triangle is a triangle with no congruent sides
  • Isosceles triangle: An isosceles triangle is a triangle with at least two congruent sides
  • Equilateral triangle: A equilateral triangle is a triangle with three congruent sides

  Because an equilateral triangle is also isosceles, all Triangles are either scalene or isosceles. But when people call a triangle isosceles, they’re usually referring to a triangle with only two equal sides, because if the triangle had three equal sides, they’d call it equilateral.

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  Mathematical Techniques in GIS

  • Peter Dale(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  59 4 The Geometry of Common Shapes 4.1 Triangles AND CIRCLES Geometry is the study of constructible shapes. In this chapter we will review some of the shapes that occur in geomatics and GIS that occupy ordinary two- or three-dimensional “Euclidian” space. Euclid was a Greek mathematician of the 3rd cen-tury bc who worked out a series of axioms or postulations concerning points, lines, angles, surfaces, and volumes. From these, he derived 465 theorems. His basic axioms included such statements as that for any two distinct points there is only one straight line that passes through them and if three distinct points are not on a straight line then there is only one plane that will pass through them. Euclid identified 10 axioms but subsequently a further one was added, namely that only one straight line can be drawn parallel to a given line through any point not on that line. In Euclidian space, the shortest distance between two points is a straight line. Two lines that are either parallel or intersect form a plane—in fact parallel lines may be said to intersect at a point at infinity, an important consideration when drawing images of three-dimensional (3D) objects in perspective on a plane (2D) surface, as discussed in Chapter 10. The triangle is the simplest shape that is made up of straight lines. In fact, all 2D shapes can be regarded as being made up from a series of Triangles, just as every curve can be thought of as a series of short straight lines. Although this can give rise to a number of errors, for example, when calculating an area enclosed by a curved line, the approximation can be adequate for many practical purposes. Triangles come in all sorts of shapes and sizes but the basic fact is that the angles of a plane triangle add up to half of a complete turn or 180 ° ; the angles of a spherical triangle, which is one drawn on the surface of a sphere, add up to more than 180 ° . For the present, we will only consider plane Triangles.

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  Triangle Geometry & Triangle Numbers

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or ver-tices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of Triangles Euler diagram of types of Triangles, using the definition that isosceles Triangles have at least 2 equal sides, i.e. equilateral Triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral Triangles isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene Triangles are also right Triangles.

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  Course in Triangle Geometry, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or ver-tices and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of Triangles Euler diagram of types of Triangles, using the definition that isosceles Triangles have at least 2 equal sides, i.e. equilateral Triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral Triangles isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene Triangles are also right Triangles.

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  Triangle Geometry & Trigonometry

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Introduction to Triangle Triangle A triangle Edges and vertices 3 Schläfli symbol {3} (for equilateral) Area various methods; see below Internal angle (degrees) 60° (for equilateral) A triangle is one of the basic shapes of geometry: a polygon with three corners or verti-ces and three sides or edges which are line segments. A triangle with vertices A , B , and C is denoted ABC . In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space). ________________________ WORLD TECHNOLOGIES ________________________ Types of Triangles Euler diagram of types of Triangles, using the definition that isosceles Triangles have at least 2 equal sides, i.e. equilateral Triangles are isosceles. By relative lengths of sides Triangles can be classified according to the relative lengths of their sides: • In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. • In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral Triangles isosceles Triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles. • In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene Triangles are also right Triangles.

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  Scientific American Science Desk Reference

  • (Author)
  • 2008(Publication Date)
  • Trade Paper Press

    (Publisher)

  similar . All circles are similar. All squares are similar. Triangles are similar if corresponding angles are equal.

  angles in Triangles The exterior angle of a triangle is equal to the sum of the opposite interior angles.

  triangle types of triangle.

  triangle A triangle is a three-sided plane figure, the sum of whose interior angles is 180°. Triangles can be classified by the relative lengths of their sides. A scalene triangle has three sides of unequal length; an isosceles triangle has at least two equal sides; an equilateral triangle has three equal sides (and three equal angles of 60°).

  median The median is the name given to a line from the vertex (corner) of a triangle to the midpoint of the opposite side.

  A right-angled triangle has one angle of 90°. If the length of one side of a triangle is l and the perpendicular distance from that side to the opposite corner is h (the height or altitude of the triangle), its area A = lh .

  hypotenuse The longest side of a right-angled triangle, opposite the right angle, is the hypotenuse. It is of particular application in the Pythagorean theorem (the square of the hypotenuse equals the sum of the squares of the other two sides), and in trigonometry where the ratios sine and cosine (see under trigonometry below) are defined as the ratios opposite/hypotenuse and adjacent/hypotenuse respectively.

  altitude The altitude of a figure is the perpendicular distance from a vertex (corner) to the base (the side opposite the vertex).

  circle Technical terms used in the geometry of the circle; the area of a circle can be seen to equal πr2 by dividing the circle into segments which form a rectangle.

  The Pythagorean theorem

  The Pythagorean theorem states that in a right-angled triangle, the area of the square on the hypotenuse (the longest side) is equal to the sum of the areas of the squares drawn on the other two sides. If the hypotenuse is h units long and the lengths of the other sides are a and b , then h 2 = a 2 + b 2

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  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The points at which the sides intersect are called vertices. A regular polygon has sides that are all the same length and angles that are all the same measure. The number of vertices of a polygon is equal to the number of sides it has. side side side side side vertex vertex vertex vertex vertex Polygon Regular polygon Number of Sides Name of Polygon Number of Sides Name of Polygon 3 triangle 8 octagon 4 quadrilateral 9 nonagon 5 pentagon 10 decagon 6 hexagon 12 dodecagon Classifying Polygons Quadrilateral (4 sides) Hexagon (6 sides) Octagon (8 sides) A triangle is a polygon with three sides (and three vertices). Triangles can be classified according to the lengths of their sides. Tick marks indicate sides that are of equal length. Equilateral triangle (all sides of equal length) Isosceles triangle (at least two sides of equal length) Scalene triangle (no sides of equal length) Triangles can be classified by their angles. Acute triangle (has three acute angles) Obtuse triangle (has an obtuse angle) Right triangle (has one right angle) The longest side of a right triangle is called the hypotenuse, and the other two sides are called legs. The hypotenuse of a right triangle is always opposite the 90° (right) angle. The legs of a right triangle are adjacent to (next to) the right angle. Right triangle Leg Leg Hypotenuse (longest side) In an isosceles triangle, the angles opposite the sides of equal length are called base angles. The third angle is called the vertex angle. The third side is called the base. Isosceles triangle theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse of the isosceles triangle theorem: If two angles of a triangle are congruent, then the sides opposite the angles have the same length, and the triangle is isosceles. Isosceles Triangles Base angle Base angle Base Vertex angle SECTION 9.3 Polygons and Triangles Copyright 2019 Cengage Learning.

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  Geometry Revisited

  • H. S. M. Coxeter, S. L. Greitzer(Authors)
  • 1967(Publication Date)
  • American Mathematical Society

    (Publisher)

  C H A P T E R 1 Points and Lines Connected with a Triangle With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more inter- esting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathe- matics. E. T. BdL The purpose of this chapter is to recall some of these half-forgotten things to which Dr. Bell referred, to derive some new theorems, developed since Euclid, and to apply our findings to interesting situations. We consider an arbitrary triangle and its most famous associated points and lines: the circumcenter, medians, centroid, angle-bisectors, incenter, excenters, altitudes, orthocenter, Euler line, and nine-point center. The angle-bisectors lead naturally to a digression on the Steiner- Lehmus theorem, which was believed for a hundred years to be difficult to prove, though we see now that it is really quite easy. Finally, from a triangle and a point P of general position, we derive a new triangle whose vertices are the feet of the perpendiculars from P to the sides of the given triangle. This idea leads to some amusing developments, some of which are postponed till the next chapter. 1.1 The extended Law of Sines The Law of Sines is one trigonometric theorem that will be used fre- quently. Unfortunately, it usually appears in texts in a truncated form that is not so useful as an extended theorem could be. We take the liberty, therefore, of proving the Law of Sines in the form that we desire. 1 2 POINTS, LINES CONNECTED WITH A TRIANGLE Figure 1.1A We start with AABC (labeled in the customary manner) and circum- scribe about it a circle with center at 0 and with radius equal to R units, as shown in Figures 1.1A and 1.1B. We draw the diameter CJ, and the chord B J.t In both of the situations shown, L CB J is a right angle, since it is inscribed in a semicircle.

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  Teaching and Learning Algebra

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  The two final problems provide a fine contrast between the two attractions of mathematics -the intrinsic interest of a paper-folding exercise which generates a 3,4, 5 triangle and the extrinsic interest of a formula that enables you to calculate how far you can see from the top of a hill. The theorem of Pythagoras provides the relationship between the lengths of the sides of right-angled Triangles, but we also need to establish relationships linking the lengths to the angles. This is the subject of Chapter 10, which discusses how students can be helped to make sense of trigonometry and to appreciate the fascinating properties and wealth of applications of the sine, cosine and tangent functions.

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  Geometry

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2004(Publication Date)
  • Wiley

    (Publisher)

  30 GEOMETRY A right triangle contains a right angle. The following triangle contains a right angle. The box shown in the following figure represents a right angle. A right angle is 90°. Next we introduce three more types of Triangles. An equilateral triangle has sides of equal lengths, and all three angles are 60 °. The following triangle is an equilateral triangle. The single line drawn across each side of the following triangle indicates that the sides are of equal length. An isosceles triangle has two sides of equal length, and the angles opposite the two equal sides have equal measurements. The following triangle is an isosceles triangle. A scalene triangle has three sides of different lengths, and all three angles have different measurements. The following triangle is a scalene triangle. Do you recall the formula (n − 2)180° = (3 − 2)180° = (1)180° = 180°? We’ll assume that the sum of the angles of any triangle is 180°. Triangles 31 Example 1: If a triangle contains angles of 29° and 58°, how much is the measure of the third angle? Solution: We’ll write an equation to show the sum of the angles is 180°. We’ll let x repre- sent the measure of the unknown angle and solve for x. 180° = 29° + 58° + x° Add 29 and 58. 180° = 87° + x° Subtract 87 from both sides of the equation. 93° = x° Example 2: Find the measure of the third angle of an isosceles triangle if its two base angles are 50°. Solution: Again we will write and solve an equation to represent the sum of the angles of the triangle. We already know the base angles of an isosceles triangle are congru- ent; in this case, both are 50°. 180° = 50° + 50° + x° Add 50 and 50. 180° = 100° + x° Subtract 100 from both sides of the equation. 80° = x° Example 3: Find the measure of the base angles of an isosceles triangle the third angle of which measures 56°. Solution: Again we’ll write and solve an equation to represent the sum of the measures of the angles of a triangle.

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  Dr. Math Presents More Geometry

  Learning Geometry is Easy! Just Ask Dr. Math

  • (Author)
  • 2005(Publication Date)
  • Jossey-Bass

    (Publisher)

  One solution (7, 7, 1) was just discussed. But even this is not quite enough. You see, (2, 5, 8) would be another solution to the equa- tion, but you couldn’t form a triangle with those lengths. With 8 as one side, the two sides of 2 and 5 (whose sum is 7) wouldn’t “meet” or connect. So we need to add one more fact to our search, called the trian- gle inequality property. This says that in any triangle, the sum of the lengths of any two sides must exceed the length of the third side. In our good example, we have 7 + 1 > 7; but in our bad example, we have 2 + 5 < 8. With a little patience, we can systematically form a list of solutions: 1. 1-7-7 2. 2-6-7 3. 3-6-6 4. 3-5-7 5. 4-4-7 6. 4-5-6 7. 5-5-5 Triangles: Properties, Congruence, and Similarity 57 2 2 Note how we let a equal the smallest side and kept it constant as long as possible while looking for the lengths of b and c. This is just to get organized. If you are concerned about obtaining all the solu- tions, it helps to have a systematic way of searching for them, such as letting a be the length of the smallest side. —Dr. Math, The Math Forum Centers of Triangles When you talk about the center of a circle, there is only one possible point. If you’re talking about centers of other objects, it’s often a bit more complicated. How do you find the center of your backyard, for example? It certainly depends on what shape the yard is! In this sec- tion, we’ll look at the different types of centers of Triangles. Hi, Quentin, To understand what facts we are given and what we need to prove, let’s review some definitions. The altitude, median, and angle bisec- tor of a triangle are all line segments that join one vertex of a trian- 58 Dr. Math Presents More Geometry Dear Dr. Math, I don’t understand the difference between angle bisectors, medians, and altitudes. Here’s a problem that I have to prove: In an isosceles triangle, the altitude is a median and an angle bisector.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A Look Back at Chapter 3 In this chapter, we considered several methods for proving Triangles congruent. We explored properties of isosceles Triangles and justi-fied construction methods of earlier chapters. Inequality relation-ships for the sides and angles of a triangle were also investigated. A Look Ahead to Chapter 4 In the next chapter, we use properties of Triangles to develop the prop-erties of quadrilaterals. We consider several special types of quadrilat-erals, including the parallelogram, kite, rhombus, and trapezoid. Key Concepts 3.1 Congruent Triangles • SSS, SAS, ASA, AAS • Included Side, Included Angle • Reflexive Property of Congruence (Identity) • Symmetric and Transitive Properties of Congruence 3.2 CPCTC • Hypotenuse and Legs of a Right Triangle • HL • Pythagorean Theorem • Square Roots Property Summary 162 CHAPTER 3 ■ Triangles In algebra, it is shown that ; not by coincidence, the set which has 0 elements, has 1 subset. Just as , the set which has 1 element, has 2 subsets. The pattern continues so that a set with 2 elements has subsets and a set with 3 elements has subsets. A quick examination suggests this fact: The entries of the fifth row of Pascal’s Triangle correspond to the numbers of subsets of the four-element set ; of course, the subsets of must have 0 elements, 1 ele-ment each, 2 elements each, 3 elements each, or 4 elements each. Based upon the preceding principle, there will be a total of subsets for . EXAMPLE 1 List all 16 subsets of the set by considering the fifth row of Pascal’s Triangle, namely 1 4 6 4 1.

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