Triangles Definition and 215 Threads

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted



△
A
B
C


{\displaystyle \triangle ABC}
.In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.

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  1. nukeman

    Similar triangles. How does 4/6 turn into 2h/3 ?

    Homework Statement r/h = 4/6 This turns into: r = 2h/3 How, can someone please refresh me? I am doing a related rates problem, and I for this! :( Homework Equations The Attempt at a Solution
  2. J

    Can You Prove the Inequality in Triangles with Arbitrary Points?

    Hi Given is a triangle on points x,y,z in the plane. This triangle has two points a and b on opposite sides (see Figure). I would like to show that the following inequality has to hold: \max {d(b,x), d(b,y), d(b,z)} + \max {d(a,x), d(a,y), d(a,z)} - d(b,a) > \min {d(x,y), d(x,z)...
  3. M

    Software to draw specific triangles?

    Hello, I am working on a project that involves using 2D triangles to build 3D models and I am looking for software that will allow me to draw 2D triangles based on the triangles 3 edge lengths (ex: 4 cm x 5 cm x 3 cm). Most of the software I have found requires knowing either the triangles...
  4. S

    Prove this inequality for all triangles

    Homework Statement Show that the angles a, b, c of each triangle satisfy this inequality. \tan \frac{a}{2}\tan \frac{b}{2} \tan \frac{c}{2} (\tan \frac{a}{2} + \tan \frac{b}{2} + \tan \frac{c}{2}) < \frac{1}{2} Homework Equations The Attempt at a Solution I used the half angle...
  5. T

    Proving # Equilateral Triangles on Sphere

    Homework Statement Ok, I have this problem this week. (1) Consider\ a\ tiling\ of\ the\ unit\ sphere\ in\ \mathbb{R}^{3}\ by\ N\ equilateral\\ triangles\ so\ that\ the\ triangles\ meet\ full\ edge\ to\ full\ edge\ (and\ vertex\ to\ vertex).\\ Show\ that\ the\ only\ possibilities\ for\ N\...
  6. C

    MHB Toothpicks in the nth figure problem where the figures are "L" shaped triangles

    Hi, I did a search in the forums and found a similar problem, but nothing on this particular one. I have a problem asking for a procedure to find the number of toothpicks in any figure in a sequence, with an illustration that shows the first three figures of the sequence of triangles built with...
  7. Z

    Are Negative Angles Important in Geometry?

    Hi everybody; I am a bit stuck with negative angles. What I know is that: If we measure an angle from x-axis clockwise direction, the angle is negative. However, I am a bit dissapointed because while I researched the web I see the negative agles only in topics subject to trigonometry. So I...
  8. P

    Question about angles and triangles I thinkg

    1. How does the moon's diameter compare with the distance between Earth and the Moon Tape a small coin, such as a dime to a window and view it with one eye so that it just blocks out the full Moon. This occurs when your eye is about 110 coin diameters away. Then the ratio of coin diameter/...
  9. A

    Why do we study right triangles in trigonometry?

    Why do we study so much about right triangles like trigonometry. We could define sin and cos like functions in a 70 degree triangle too. I also know right triangle is something special but i don't know what is it. Also why won't trigonometry on other type of triangles be not so good
  10. S

    Can a Right Triangle's Hypotenuse be Found from Its Area and Perimeter?

    A Right-Angled Triangle has area Acm^2 and perimeter Pcm. A side other than the hypotenuse has length has length xcm. Form a quadratic equation in x in each of the following cases: a) a=6 p=12 let the other side be y, and the hypotenuse be h x + y + h = 12 0.5*y*x = 6, y= 12/x x + 12/x...
  11. A

    Triangles and Definite Integrals

    I'm trying to figure out how to integrate a data set, without knowing the function. While doing this, I got to thinking about this: If the definite integral of a function can be represented by the area under that function, bound by the x axis, then shouldn't: \int_{a}^{b}2x\frac{\mathrm{d}...
  12. V

    Forces-vector diagram triangles

    when a cable car is stationary,it is in equilibrium.So the forces acting on it forms a closed triangle.(the vector diagram) Why is the triangle an isosceles triangle? I did this question on my paper and stumbled across this on the answers,couldnt figure out why hope someone could enlighten me!
  13. N

    Estimating 3d transformation of two triangles

    I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D. 1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the origin(to get rid of the translation part)? 2)How does it affect if the deformation is non rigid?
  14. Solarmew

    Proving At Least 3 Pure Triangles in a Complete Heptagon

    errrr, THEOREM >.< ... oops ... Homework Statement Prove that when edges of a complete heptagon are colored with two different colors, there will be at least three pure triangles.Homework Equations The Attempt at a Solution i can do two pure triangles, but not three :cry: pick a vertex v. It...
  15. C

    Trignometry to non-right angled triangles?

    I learned trignometry on the basis of right-angled triangles.Is it applicable to all(acute and obtuse) triangles?and how do you explian cos or sin(<=90),since it's no longer associated with right angled triangle?.can anyone please clarify.
  16. G

    Optimization of a Triangles Area.

    Hello everyone, This is my first post on here, I was hoping it wouldn't be asking for help but I don't have any options left and it's a problem that is due soon. I promise I have tried to solve it myself but I'm unsure if I'm doing it correctly and it is part of a take home test. Homework...
  17. S

    Equivalent Resistance: Series or Parallel Triangles?

    Homework Statement Okay. I thought I knew how to do these type of questions, but here goes. The node at a is attached to the positive terminal of a voltage source and c is attached to the negative. I'm completely bemused as to how to reduce this circuit into a single-resistor equivalent...
  18. M

    Finding a side length in similar triangles?

    Side note: PF is awesome! This is kind of like Tutor.com, only it's free and others can contribute their answers. Homework Statement Explain why triangle ABD is similar to triangle ABC and then find the length of side AB. Angles B and C are congruent. m(ADB)+m(BDC) = 180 degrees. AD=4 and...
  19. N

    What is center of gravity espcially for triangles

    What is ceter of gravity concept and how to calculate it for a triangle
  20. N

    Noob to PF, Quantum Foam question triangles?

    Hi I am a noob on here, first post. Not a physicist by any stretch, but am researching the quantum mind and a while ago came across a paper that proposed that the quantum foam (if that is the correct term, maybe spacetime?) was geometrically just a bunch of triangles. If anyone can post a link...
  21. Z

    Comparing Line Segments in Triangles

    Homework Statement So these are line segments in triangles. I don't understand how they are different. Homework Equations The Attempt at a Solution
  22. R

    Who is Behind the Mysterious Flying Triangles?

    http://www.ufoevidence.org/Cases/CaseView.asp?section=Triangle http://www.ufoevidence.org/topics/triangles.htm They have been sighted all over the world including a friend of mine who saw it up close just hovering about 4 storey high in a 2 storey home almost covering it. Who created them...
  23. J

    Graphs: Expected Number of Triangles and Variance

    Homework Statement Let G be a random graph on n vertices: 1) What is the expected number of triangles in G? 2) What is the variance in the number of triangles? Homework Equations N/A The Attempt at a Solution I think I can do (1) by using indicator variables. In particular, let...
  24. R

    Pythagorean Triangles with one side equal s and hypothenuse equal 2 s+1

    This thread is concerned with rectangular triangle with one side ( = s) and hypothenuse (t = 2 s + 1) given and we are looking for the remaining side (all sides having integer values) d^{2} = t^{2} - s^{2} = (2 s + 1)^{2} - s^{2}, with d, s, t \in \mathbb{Z} Computational experiments...
  25. N

    I thought it was similar triangles

    Homework Statement http://i54.tinypic.com/2cdib6t.jpg Since its trig, here is a scan. Homework Equations The Attempt at a Solution I thought it was similar triangles, so i used ratio to solve it as 10.4. what if it asks for me ar angle instead of a side? is that...
  26. E

    Algebraic expression for triangles within polygons

    Homework Statement If a polygon has n>=4 sides, what is the probability, in terms of n, that a triangle made up of vertices of the polygon shares at least one side with the polygon. Homework Equations The Attempt at a Solution Treating vertices of polygons as possible outcomes...
  27. D

    Relationship Between Triangles and Straight Angles

    My girlfriend is re-learning geometry. She has noticed that the sum of the angles of a triangle is equal to the measure of a straight angle and half the measure of a full revolution. She wants me to ask if there is any special relationship between a triangle and a straight angle or between a...
  28. agnibho

    Problem in similarity of triangles

    Homework Statement ABCD is a parallelogram. E, F are points on the straight line parallel to AB. AF, BF meet at P, and DE, CF meet at Q. Prove that PQ ll AD. 2. The attempt at a solution I drew the diagram. I tried to solve the problem in this way:- CD ll XY Therefore, angleCDE = angleDEF...
  29. M

    Circles and Triangles: Solving for X with Geometry

    X = ?
  30. R

    Conjecture: Prime Divisibility & First Differences of Stirling & Eulerian Triangles

    CONJECTURE: Subtract the Absolute Values of the Stirling Triangle (of the first kind) from those of the Eulerian Triangle. When row number is equal to one less than a prime number, then all entries in that row are divisible by that prime number. Take for instance, row 6 (see below). The...
  31. S

    Using the Cosine Law on Right and Oblique Triangles

    Can the cosine law be used on right triangles as well as oblique triangles?
  32. H

    Similar Triangles, Light and Shadow.

    Homework Statement A man 6 feet tall walks at a rate of 5 feet per second away from a light that is attached to a pole 15 feet above the ground. At what rate is the length of his shadow changing when he is 30 feet from the base of the pole? I get that this is really like a two similar...
  33. agnibho

    Areas of Parallelograms and Triangles

    Homework Statement ABC and BDE are two equilateral triangles such that D is the midpoint of BC. If AE intersects BC at F, show that :- (i) ar(BDE) = 1/4 ar(ABC) (ii) ar(BDE) = 1/2 ar(BAE) (iii) ar(ABC) = 2ar(BEC) (iv) ar(BFE) = ar(AFD) (v) ar(BFE) = 2ar(FED) (vi) ar(FED) = 1/2ar(AFC)...
  34. D

    Geometry Homework: Sum of Circles and Triangles in Figure

    Homework Statement Please see the attached figure The radius of the biggest circle is 10. The required is the sum of all circles and the sum of all triangles in the figure. There is an infinite number of circles and triangles. My answers are: for circle: 475/3 pi for triangle: 175/2...
  35. C

    Trigonometric Substitution Triangles

    Homework Statement This may be basic, but how do you know which part of an integration problem fill the up the triangle values for trig substitution?
  36. M

    Can any triangle use the Pythagorean theorem?

    I am reading about "special triangles" 45.45.right angle. I was reading how they found the ratios of the sides of a triangle. It looks like they let the leg opposite of the 45 deg angle be length A and the corresponding 45 deg angle be length A aswell.. then they used the Pythagorean theorem...
  37. S

    Why Isn't the Hypotenuse the Sum of the Other Two Sides?

    Lately this has been bothering me, I hope you can understand the point I'm trying to make. On a side note, this topic maybe more philosophical than mathematical. It comes down to this. In a right triangle, why is the length of the hypotenuse not equal to the length of the adjacent side plus the...
  38. R

    Conjecture: Sophie Germain Triangles & x | 2y^2 + 2y - 3 = z^2

    Limiting ourselves to N... Conjecture: (sqrt (16x + 9) - 1)/2 = y | 2y^2 + 2y - 3 = z^2 for x = a Sophie Germain Triangular Number, which is recursively defined as: a(n)=34a(n-2)-a(n-4)+11 First 11 values (I have not checked further as I would be very surprised if this equivalency...
  39. T

    Geometry with circles, triangles and squares

    Homework Statement The diagram below shows four congruent circles whose centres are the vertices of the square DEFG and whose circumference touch the sides of an isosceles triangle. Area of triangle ABC is 10000 units square. What is the radius of the circles, to the nearest unit...
  40. D

    Physics of Triangles: Why Are They the Strongest?

    I have always heard that triangles are the strongest geometric shapes. Can anyone explain why this is so?
  41. I

    Pasch's Theorem for Asymptotic Triangles

    Theorem: If a straight line intersects one of the sides of the asymptotic triangle ABOmega but does not pass through a vertex (including omega) it will intersect exactly one of the other two sides. Proof. I have a few ideas for this proof. First, I think it will be three cases where the...
  42. C

    Proving triangles with vector methods

    Homework Statement In the following diagram D and E are the midpoints of AB and AC. Use vector methods to prove that DE = 1/2BC Homework Equations DE = 1/2 BC The Attempt at a Solution AD = 1/2AB AE = 1/2AC AD + DE = AE DE = -AE + AD DE = -1/2AC + 1/2AD IF BC = -AC + AB AND DE = -1/2AC +...
  43. O

    Can a graph with no triangles have more than (n^2)/4 edges?

    [b]1. Prove that a graph with n vertices that has no triangles has at most (n^2)/4 edges. [b]2. [b]3. Um so i was thinking maybe you start off with a complete graph and then keep deleting edges so as to remove the triangles within it or something? Or maybe by induction?
  44. S

    Prove Existence of 5 & 64 Points in Plane with 8 & 2005 Right-Angled Triangles

    Prove that there exist (a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones; (b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.
  45. A

    Find the constant unknown angle of triangles series

    http://a.imageshack.us/img715/8526/paint2.jpg there are 2 triangles. in these triangles r1,a,theta1 r2,a,theta2 are given. x1 and x2 are unknown. also the angle k is same for both triangles. i need to find this k. also i have data to draw series of tingles like these 2(i know r3,theta3,a...
  46. B

    Geometry with isosceles triangles

    2 isosceles triangles ABC and ABD have the same base AB but are in different planes. Prove that CD is perpendicular to AB. (It must be proved that CD is perpendicular to one of the planes and then it is perpendicular with every segment of this plane.) Thanks in advance!
  47. P

    For each positive integer n, let T(n) be the number of triangles with

    Homework Statement For each positive integer n, let T(n) be the number of triangles with integer side lengths, positive area, and perimeter n. For example, T(6) = 1 since the only such triangle with a perimeter of 6 has side lengths 2, 2 and 2. (a) Determine the values of T(10), T(11) and...
  48. D

    Proportionality of triangles problem

    Homework Statement Is any formulas similar to that of proportionality of the triangles to proportionality of the given figure? Homework Equations The Attempt at a Solution
  49. R

    How Are Triangles Used to Represent Vectors in Physics?

    As someone who has just started Physics, when triangles were introduced in my physics it caught me in surprise. If the legs of triangle were heights It would somewhat make more sense to me, but our teacher says their are velocities. I do not understand. For example : A pitched ball is...
  50. R

    Why are spheres and triangles so important?

    I've notice that our eye balls are spheres, our Earth is a sphere and our sun is a sphere.. Now that makes me wonder. If our universe is a sphere, i already know scientist think so, but its still something to even think more into. Now this is the big part, triangles are important as well. Like a...
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