What is Triangle: Definition and 1000 Discussions

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. GreenGoblin

    MHB Max Area of isosceles triangle with perimeter 1

    Given this seemingly simple problem of maximising the area of an isosceles triangle with perimeter equal to 1. What is the best approach and how will I find a result easiest (I know how to get the answer but I need to be able to do these problems fast, so please help me look for a quick method...
  2. B

    Related Rates problem involving triangle

    Homework Statement "At a given instant the legs of a right triangle are 8in. and 6in., respectively. The first leg decreases at 1in/min and the second increases at 2in/min. At what rate is the area increasing after 2 minutes?"Homework Equations A=\frac{1}{2}bh \frac{db}{dt}=-1...
  3. anemone

    MHB In a triangle ABC, prove that 1<cosA+cosB+cosC< or equal to 3/2

    In a triangle ABC, prove that $1<cosA+cosB+cosC \leq \frac{3}{2} $. One can easily prove that $cosA+cosB+cosC \leq 3/2 $, i.e. it can be proven to be true by 1. Using only the method of completing the square with no involvement of any inequality formula like Jensen's, AM-GM, etc. 2. By...
  4. anemone

    MHB Can x, y, and z be the side lengths of a triangle?

    Let ABC be a triangle. Prove that $sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=1$. Conversely, prove that if x, y and z are positive real numbers such that $x^2y^2+z^2+2xyz=1$, then there is a triangle ABC such that $x=sin\frac{A}{2}...
  5. B

    Discovering the Length of a Triangle in Thales' Time

    Homework Statement The Attempt at a Solution I can't figure out how thales found the measurement of the length of this triangle without trigonometry since the sine and cosine ratios were not worked out until the 14th century I think. In any case, they certainly weren't known in...
  6. C

    Calculate Height of Triangle: Pythagoras Formula & Ratio

    Homework Statement Please see picture attached. Homework Equations The Attempt at a Solution If I use the pythagoras formula with the ratio I get the height as 3 yet it is longer than the side that is 4. Am I doing something wrong? Thanks.
  7. D

    MHB Triangle inequality question

    I am trying to show $|(n+z)^2|\leq (n -|z|)^2$ where is complex $|(n+z)^2| = |n^2 + 2nz + z^2| \leq n^2 + 2n|z| + |z|^2$ But I can't figure out the connection for the final piece.
  8. S

    Pascals Triangle, arithmetic sequence.

    Homework Statement Write down the condition for the numbers p, q, r to form an arithmetic sequence. Homework Equations The Attempt at a Solution Have no idea, but I looked at the answer and they have assigned each letter with a given value (number). How is this possible?
  9. B

    Another Isosceles Triangle Problem

    Homework Statement In isosceles triangle RST, RS = RT. Which side of the triangle is the base? Which angle is the vertex angle? Homework Equations The Attempt at a Solution I believe the base is RT, and the vertex angle is S. I don't know the proper notation to name the line...
  10. Advent

    Triangle inequality for complex numbers: sketch of proof

    Homework Statement Show that if z_1,z_2 \in \mathbb{C} then |z_1+z_2| \leq |z_1| + |z_2| Homework Equations Above. The Attempt at a Solution I tried by explicit calculation, with obvious notation for a,b and c: my frist claim is not that the triangle inequality holds, just that...
  11. B

    Isosceles Triangle Geometry Problem

    Homework Statement The measure of the sides of an isosceles triangle are represented by x + 5, 3x +13, and 4x + 11. What are the measures of the sides? Two answers are possible. Homework Equations The Attempt at a Solution Well, I set up three different triangles, to account for the...
  12. P

    Probability to get a specific triangle

    Three points are randomly chosen on a circle. Then they are connected together to make a triangle. What's the probability to see a 1. Right triangle 2. Triangle with all angles less than 90 degrees 3. Triangle with an angle bigger than 90 degrees I'm pretty sure to solve one of them...
  13. C

    Calculating the Width of a Triangle at Position x

    Homework Statement Given an isosceles triangle - Length = L - Uniform Density = ρ - Width Varies from 0 at x = 0 to a at x = L I attached a picture of it. Homework Equations Have to show the width at position x is given by (a/L)x The Attempt at a Solution Now it is...
  14. J

    Drawing a right triangle to simpliy the given expressions

    1. Homework Statement Sin(sec^-1(sqrt(x^2+16)/4)) 2. Homework Equations 3. The Attempt at a Solution I did the math and ended up getting x^2-1 as the opposite, but the answers on the back of the book say other wise.
  15. J

    Drawing a right triangle to simply the given expressions

    Homework Statement Cos(Tan^-1(x)) Homework Equations The Attempt at a Solution Dont know how drawing a right triangle would do anything, don't know what to plug in.
  16. V

    Triangle integral ∫∫dxdyf(x*y) how to reduce to one dimension?

    I meet with a triangle integral where x+y≤1, and function is dependant only on x*y. I am wondering if there any possibility to relate ∫∫dxdyf(x*y)=∫d(x*y)f(x*y)g(x*y) or something similar? Or maybe there are some assumptions needed to relate like this?
  17. H

    Striking the End of a Triangle in Space

    Homework Statement http://img221.imageshack.us/img221/4861/figure1h.png Mod note: Fixed the image link. Consider the rigid object shown in this image. Four masses lie at the points shown on a rigid isosceles right triangle with hypotenuse length 4a. The mass at the right angle is 3m...
  18. C

    Electric Forces in a Triangle (math check?)

    Hi Guys I've tried to do this problem but the answers are not being accepted is the math wrong? or did i forget a concept? Thanks for taking a look The triangle is set by Q1 on the top of the triangle Q2 on the bottom left and Q3 on the bottom right Calculate the magnitude of the net...
  19. L

    Geometry Triangle Congruence - Should be easy?

    Geometry Triangle Congruence - **Edited to include diagram** Homework Statement Hi! The three simple geometry problems are in the attached photo. Sorry if they're difficult to read. I haven't seen this information in so long, and could use some help. :D Can you use the given information...
  20. xortan

    Slope and displacement of a triangle beam

    Homework Statement I have attached a picture of the problem. Homework Equations Singularity functions? The Attempt at a Solution So I am an electrical engineering student and got this mechanics course this semester and the teacher hasnt really explained anything, its almost like...
  21. F

    MHB Perpendicular vectors, triangle, tetrahedron

    Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results: a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point. b. If the tetrahedron OABC has two pairs of...
  22. F

    Perpendicular vectors, triangle, tetrahedron

    Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results: a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point. b. If the tetrahedron OABC has two pairs of...
  23. S

    Area of a Triangle Word Problem

    Homework Statement If the height of a triangle is five centimeters less than the length of its base and if the area of the triangle is 52 cm^2, find the base and the height.Homework Equations N/A The Attempt at a Solution height = h-5 base = h A=(bh)/2 52=(h(h-5)/2 52=(h^2-5h)/2 104=h^2-5h...
  24. L

    Electric Fields in Equilateral Triangle

    Homework Statement A point charge q = +2.0 µC is placed at each corner of an equilateral triangle with sides 0.23 m in length. What is the magnitude of the electric field at the midpoint of any of the three sides of the triangle? 3. I want to know if I did this correctly, or totally...
  25. B

    Triangle and dot product to find angle

    Vector A and vector B are expressed in component form. A = [2.32,-5.16,7.88] B = [-1.12,3.45,-12.8] The standard arrow representation of these vectors and that of can be arranged to form a triangle in a plane that represents the geometric equivalent of the subtraction operation. The...
  26. T

    Triangle Inequality for a Metric

    Homework Statement Prove the triangle inequality for the following metric d d\big((x_1, x_2), (y_1, y_2)\big) = \begin{cases} |x_2| + |y_2| + |x_1 - y_1| & \text{if } x_1 \neq y_1 \\ |x_2 - y_2| & \text{if } x_1 = y_1 \end{cases}, where x_1, x_2, y_1, y_2 \in \mathbb{R}...
  27. S

    Is the interval the height of the triangle?

    A FAQ is why the spacetime interval has a minus sign. A geometrical answer is that the interval is always the shortest path (the straight line) between two points. If we talk about spatial points, that interval is the hypotenuse of a right angle triangle, whose sides are the projections of...
  28. T

    Using Triangle Inequality to find a magnitude

    Homework Statement Let a, x, and y be real numbers and let E > 0. Suppose that |x-a|< E and |y-a|< E. Use the Triangle Inequality to find an estimate for the magnitude |x-y|. Homework Equations The Triangle Inequality states that |a+b| <= |a| + |b| is valid for all real numbers a and...
  29. B

    Area of a circle within a circumscribed triangle

    If you have a triangle circumscribed around a circle, how do you find the area of that circle? Say that the triangle is an equilateral triangle with side length of 8 cm. I found the area of the triangle using Heron's formula: 16√3 cm^2. Apparently the answer is 16π/3 cm^2. I'm just confused...
  30. P

    MHB Area of an equilateral triangle

    Given an equilateral triangle ABC, P is any point inside it where PA = 3, PB = 4 and PC = 5. Find area of the triangle using the Law of Sines or Law of Cosines.
  31. D

    Addition of vectors with triangle vertices

    Homework Statement If A, B, and C are the vertices of a triangle, find the following. (A→B) + (B→C) + (C→A) The answer has to be given like this: _____ i + _____ j 2. Homework Equations (I think?) a+b=b+a a+(b+c)=(a+b)+c a+0=a a+(-a)=0 c(a+b)=ca+cb (c+d)a=ca+da (cd)a=c(da)...
  32. V

    Angled triangle ABC are x-1, x

    Homework Statement The length of the sides of the acute angled triangle ABC are x-1, x and x+1. BD is perpendicular to AC. Then CD-DA equals? Homework Equations cosine law The Attempt at a Solution
  33. J

    Area of Isosceles Triangle Problem

    Homework Statement An Isosceles triangle has two equal sides of length 10cm. Let x be the angle between the two equal sides. a. Express the area A of the triangle as a function of x in radians. b. Suppose that x is increasing at the rate of 10 degrees per minute. How fast is A changing...
  34. J

    Relative Error of Right Triangle Area

    Homework Statement Anyone mind checking my answer for me? The area of a right triangle with hypotenuse H is A=1/4H^2sin(2x) x is one of the acute angles. Use differentials to estimate the relative errors of the area if H = 4cm exactly and x is measured to be 30degrees with an error of...
  35. Z

    Double integral over triangle with known nodes

    I am wondering is there any general analytic solutions to the following integrals First moment of area Integral(xdxdy) Integral(ydxdy) Second moment of area) Integral(x^2dxdy) Integral(y^2dxdy) Integral(xydxdy) Over the triangle (x1,y1) (x2,y2) (x3,y3) thanks Wei
  36. B

    Triangle wave op-amp generator

    Well another op-amp thread, while the other one is being cooked. http://pokit.org/get/a09120280ee2f6b6e3de7a99989b8bd9.jpg To speed you up, this is a triangle wave op amp generator. It generates a triangle wave at U2. In general I understand how the circuit works, but from the engineering...
  37. C

    Moment of inertia of triangle rotated about y axis

    Similar to my other thread https://www.physicsforums.com/showthread.php?t=557750 I think in this case I need to use triple integrals But what is the integrand and limits? Can someone please help me set it up? Here's all I know ∫∫∫dydx Please help I have my final next wednesday.
  38. L

    Triangle Angles: Find b & c with A-B-C given

    I am stuck on the job; I need to find the angles. It’s not an angle or side based triangle, just a right triangle. The lengths are known, A-B-C. What I need to know is the angles, a-b-c. So, a=90*, what formula could I use to find b, and c? Sorry, please make it easy enough for me to work it...
  39. P

    Understanding x(θ) and h(θ) in a Triangle Support Beam Problem

    Homework Statement I have attached the problem. I am confused about how they got x(θ) and h(θ). From the diagram it looks like they made it with reference to the triangular support beam. Why? Homework Equations The Attempt at a Solution
  40. J

    How to find the maximum area of a triangle?

    Homework Statement The hypotenuse of a right triangle has length of 10cm. Determine its maximum possible area. c= 10cm Homework Equations a^2 + b^2 = c^2 ? b*h*(1/2) The Attempt at a Solution just drew out the triangle so far, confused as to where to go from here i assume ill...
  41. S

    Inequality with Circle and Triangle in Euclidean Geometry

    Homework Statement Please see below... Homework Equations Please see below... The Attempt at a Solution Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution... Thank you...
  42. J

    Center of mass of a triangle

    Homework Statement Find the center of mass of a triangle with two equal sides of length a. The triangle's third side is length b and it has a uniform mass of M. Homework Equations R = \frac{1}{M} \int dm \vec{r} dm = \frac{M}{A} A = \frac{1}{2}base*height The Attempt at a Solution...
  43. S

    Finding centre of mass of an isosceles triangle

    Homework Statement Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle. The monument is made from tens of thousands of small stone blocks of density 3 800 kg/m3. The monument is 15.7 m high and 64.8 m wide at its base and is everywhere 3.60 m thick...
  44. I

    Question 26.5: Triangle Circuit - Is My Drawing Correct?

    I am not sure how the configuration would look like if we add a voltage across these points. I have drawn out what I think it would look like after a potential is applied across Vab. My question is, is this drawing correct?? What about Vbc and Vac? This is question 26.5. And here is my...
  45. S

    Finding C_2 for Curve F in Triangle C

    Homework Statement F=<xy, x-y> and C is the triangle joining (1,0), (0,1) and (-1,0) in the clockwise direction. Homework Equations How do I have the second part of this curve? The Attempt at a Solution Apparently, there should only be a sum of two integrals. So I got that curve...
  46. J

    Where is the center of mass of a flat triangular plate with given dimensions?

    Find the x and y coordinates of the center of mass of a flat triangular plate of a height H =17.3cm and a base B = 10 Cm Homework Equations Area = A = 1/2bh x = 1/A ∫ x(f(x)) y = 1/A ∫ 1/2((f(x)^2) The Attempt at a Solution Area = 1/2(10)(17.3) = 86.5 f(x) = 17.3-1.73x or...
  47. shinnsohai

    ABC is an equilateral triangle

    Homework Statement Based on my understanding of the question the diagram should be like this ABC is an equilateral triangle with side 2a.Two circle are are drawn , with one of them crossing the points A,B and C.another circle crossess B and C with A as centre Prove that the area enclosed by...
  48. S

    Calculating Distance Between Inner & Outer Triangles

    Hi, I have an equilateral triangle and I put another equilateral triangle within it. The inner triangle has exactly half of the area of the outer triangle. As well, the inner triangle is placed in the centre of the outer triangle i.e. they both have the same central point. I want to...
  49. V

    Equilibrium and principle of triangle of forces

    1. Homework Statement A rope of 5m is fastened to two hooks 4m apart on a horizontal ceiling. To the rope is attached 10kg mass so that the segment of the rope are 3m and 2m.Compute the tension in the string 2. Homework Equations Sine rule, cosine rule 3. The Attempt at a Solution...
  50. F

    Calculating Probability of 2 Black Balls from 3-Ball Urn w/ Pascal Triangle

    if we toss 4 coins and we want to know the probability of having 3 heads we can easily calculate that using pascal triangle . I want to know if we have an Urn that contain 3 balls : 1 black , 1 red, 1 yellow. and we want to draw 3 balls , but when we draw the first ball we put it back in the...
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