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. F

    Moment of Inertia of a Right Triangle

    Homework Statement A metal sign for a car dealership is a thin, uniform right triangle with base length b and height h. The sign has mass m. What is the moment of inertia of the sign for rotation about the side of length h? Homework Equations I = \intr^{2}dm The Attempt...
  2. D

    Cauchy/ Triangle Inequality

    I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?
  3. W

    How can I find angle A in a right triangle without using a calculator?

    Homework Statement a right triangle. hypotenuse is 10, the opposite is 8, and the adjacent is 6 Sine .6 Cosine .8 Tangent .75 i need to find angle A (aka Alpha in the problem) With out using a graphing or scientific calculator. Homework Equations i was told to use...
  4. P

    Maximizing Intercepted Lengths in a Right Triangle Inscribed in a Circle

    Homework Statement A right angle is moved along the diameter of a circle of radius a (see diagram). What is the greatest possible length (A+B) intercepted on it by the circle. Homework Equations so, the pythagorean theorem might be useful diameter = 2a The Attempt at a Solution...
  5. T

    Induction proof of harmonic triangle formula

    Hello. In the Wiki entry for "Leibniz harmonic triangle" there one can find a formula L(r, c) = 1/(c * binom(r - 1, c - 1)) where L(r, c) is the entry in the harmonic triangle and the binom() is the usual binomial notation (or the entry from Pascal triangle). I tried to prove the...
  6. A

    Impossible Bearing and Triangle word problem.

    Homework Statement Radio direction finders are set up at points A and B, 8.68 mi apart on an east-west line. From A it is found that the bearing of a signal from a transmitter is N54.3*E, while from B it is N35.7*W. Find the distance from B, to the nearest hundredth of a mile. A person...
  7. mnb96

    Triangle inequaility for sets of same cardinality

    Hello, given a set \Omega, we consider all its subsets A_1,A_2,A_3,\ldots with same cardinality k. Do you have some hint in order to prove the following: \forall A_x,A_y,A_z\subseteq \Omega such that |A_x|=|A_y|=|A_z|=k |A_x-A_z| \leq |A_x-A_y|+ |A_y-A_z| Thanks
  8. B

    Triangle inscribed within a circle

    ABC inscribed within a circle whose diameter AC forms one of the sides of hte triangle. If Arc BC on the circle subtends an angle of 40 ddegrees, find the measure of angle BCA within the triangle
  9. C

    Extension of the Triangle Inequality

    Pretty much knows the triangle inequality. \left| a + b \right| \le \left| a \right| + \left| b \right| I was reading a source which asserted the following extension of the triangle inequality: \left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right) This is...
  10. P

    Calculate coordinate in 3D triangle.

    Hi, I'm looking for an efficient way of achieving the following, but having trouble thinking it through: Let's say I have a triangle in three dimensions with vertices A, B and C. Let's say I also have a three dimensional coordinate D. I know the x and z component values for D and I am...
  11. A

    Proving Convergence of Xn: Triangle Inequality & Examples

    hi every body show if Xn→x then lXnl→lxl hint use trangle inequality 2/ show if lXnl→0 then Xn →0 show by example that lXnl fore all n in N MAY CONVERGE and Xn may not converge
  12. J

    Triangle Inequality and Cauchy Inequality Proofs

    Homework Statement The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality. This is an elementary linear algebra class I'm doing, so I can't use inner products or anything. Homework Equations The Attempt at a Solution I got the proofs using algebra, but I'm...
  13. P

    Object moving in equilateral triangle - magnitude of acceleration

    Homework Statement An object moves around a equilateral triangle with a constant speed (magnitude of velocity is constant) of 22.3 m/s. If it moves from the starting to ending point (as shown below) in a time of 12.69 seconds, what is the magnitude of the average acceleration vector in m/s2...
  14. T

    Area of a non-right angled triangle with information missing

    The area of the triangle in the attachment has to be found. It is not stated in the question what kind of a triangle it is nor are there any angles given. I believe that there is information that has been left out of the diagram to solve this question (This is a question in a maths...
  15. E

    Rate of change in an Isosceles triangle

    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 at the instant x =...
  16. L

    Triangle Inequality Proof

    Homework Statement Given:0 <= a <= b a <= Sqrt(ab) <= (a+b)/2 <= b Homework Equations The Attempt at a Solution The only problem I am having prooving this inequality is Sqrt(ab) <= (a+b)/2. I have an idea but I am not sure if it validates. can i do this.. ? (a+b)/2 - sqrt(ab) >= 0 if it is...
  17. L

    Triangle Inequality Proof help

    Homework Statement 1] l x + y l < or equal to l x l + l y l Homework Equations x^2 + 2xy + y^2 The Attempt at a Solution Left side. i Squared left side to begin with, and i got x^2 + 2xy + y^2 and also did the same for the right side, but it would have absolute sign...
  18. J

    UK MOD explanation for black triangle UFO's

    UK Ministry of Defense Report Findings "The UK Ministry of Defense report "UAP in the UK Air Defence Region[2] released to the public in 2006 draws several conclusions as to the origin of "Black Triangle" UFO sightings. Their researchers conclude that "Black Triangle" UFOs are formations of...
  19. B

    Finding radius of circle inscribed in a triangle

    Homework Statement Someone gave me this problem: finding the radius of the circle inscribed in the triangle with the given lengths. The Attempt at a Solution The person asking about this problem said it was taken from a beginning algebra textbook. I tried figuring it out using just...
  20. D

    Relations between the sides of a triangle, and everythign is underroot

    The relation to be proved looks pretty simple. However, there is no evident point from whih I can start. How do I start relating quantities that are under root? The cyclic order of the sides on the left side is reminiscent of the cosine rule, but that's just it! Plus the right hand side...
  21. V

    Rate of change of a right triangle

    Imagine a boat on the water being putted towards the shore by a winch mounted a height h above the water. The winch reels in cable (shortening the hypotenuse) at constant rate u in m/s . When the boat is a distance d from the shore, find its speed and acceleration. This is an example of motion...
  22. M

    Triangle inequality w/ Complex Numbers

    given z, w\inC, and |z|=([conjugate of z]z)1/2 , prove ||z|-|w|| \leq |z-w| \leq |z|+|w| I squared all three terms and ended up with : -2|z||w| \leq |-2zw| \leq 2|z||w| I know this leaves the right 2 equal to each other but i figured if i show that since there exists a z\geqw\geq0, then...
  23. M

    Triangle inequality in Rubins book

    My problem states: Given z, w\inC, prove: ||z|-|w||\leq|z-w|\leq|z|+|w|. Now, I am confused because, isn't it true that ||z|-|w||=|z-w| ? I am using Rudin's book which gives |z|=([z's conjugate]z)1/2
  24. B

    Electric Field at the Center of a Triangle

    Homework Statement Given an equilateral triangle with sides 20-cm, with point charges at each point: q1 at the top with a charge of +4.0uM, q2 at the bottom right with a charge of -4.0uM, and q3 at the bottom left with a charge of +4.0uM. k is assumed to be exactly 9.00x109 N-m2/c2...
  25. R

    How Do You Construct a Triangle of Forces?

    What's a "triangle of forces" (triforce, lol) I've attached a snip from my textbook. I don't get how they make that triangle, and how they get those angles. Can someone please help.
  26. B

    What is the Correct Formula for the Area of an Isosceles Triangle?

    Homework Statement Imagine an isoscoles triangle. The "top" vertex has an angle of 2*theta. The common side has a length of L. When you cut the isoscoles triangle in half, that common side becomes the hypotenuse of the two resulting right-triangles. {\rm{BaseLength}} = L \cdot \sin \theta +...
  27. J

    Using pascal's triangle and standard form

    Homework Statement If z = 4 + i and w = -2 +2i, determine (z+w)5 using Pascal’s Triangle and standard form. Homework Equations The Attempt at a Solution
  28. P

    Total electric force on a particle in an equilatteral triangle

    Three charged particles are located at the corners of an equilateral triangle. Calculate the total electric force on the 7.00µC charge. The three charges are: 7.00µC, 2.00µC, and -4.00µC and they are all 0.500 meters apart. Here's a graphic of the layout...
  29. A

    Calculate Area of Triangle in 3D with Given Vertices | Homework Solution

    Homework Statement The vertices of triangle are (2,0,0) ; (0,3,0) ; (0,0,6) . Find the area of triangle. Homework Equations The Attempt at a Solution is it right that i put in determinant like Area of triangle = 0.5 | 2 0 0 | | 0 3 0 | | 0 0 6 | = 0.5 (2* 3 * 6)...
  30. H

    Checking Triangle Inequality for List Similarity Metric

    Say I have two lists, List1 and List2 containing elements such as words. Some words are common two both List1 and List2. I want to create a distance metric that tells me how far apart the two lists are based on a similarity "score". The similarity score and distance metric are as follows...
  31. P

    Measure of the smallest angle in the triangle

    Homework Statement In a triangle, two sides that measure 6 centimeters and 10 centimeters form an angle that measures 80 degrees. Find to the nearest degree, the measure of the smallest angle in the triangle. Homework Equations A2 = B2 + C2 – 2bc Cos A The Attempt at a Solution...
  32. F

    Projection of a triangle in XY plane

    Homework Statement triangle in the plane z=1/2y with vertices (2,0,0) (0,2,1) (0,0,0) please help me to find out the projection of the triangle in xy plane. thanks Homework Equations The Attempt at a Solution
  33. I

    Rectangle inscribed in Triangle

    Let PQRS be a rectangle inscribed in a triangle ABC(i.e P is in AC, Q in BC and R,S are in AB). Find the locus of points that are intersection of diagonals of the rectangle. (i.e find the locus of intersection of RQ and PS)
  34. E

    Green's Theorem and a triangle

    Homework Statement Use Green's Theorem to evaluate ∫F*dr. (Check the orientation of the curve before you applying the theorem.) F(x,y)=<y2cos(x), x2+2ysin(x)> C is the triangle from (0,0) to (2,6) to (2,0) to (0,0) *=dot product Homework Equations Green's Theorem The Attempt...
  35. S

    How to determine if a point is within a triangle?

    Homework Statement Hi, How to check whether a give point is inside or outside the triangle, Please help me . Homework Equations The Attempt at a Solution
  36. S

    Minimizing a vector in a triangle

    Homework Statement A truck is to be towed using two ropes. If the resultant force is to be 950N, directed along the positive x axis, determine the magnitudes of forces F1 and F2 acting on each rope and the angle at which rope two pulls to ensure that the magnitude of F2 is a minimum. F1 acts...
  37. S

    Finding Mass of Block B in Triangle with Pulley & Friction Coefficients

    I have a conceptual problem regarding the following scenario: Try to imagine a triangle with two blocks on each side, attached together with a rope, and a pulley at the tip of the triangle. Block A & B are attached to each other through a rope. Blocks A & B are in static equilibrium, each on...
  38. H

    Proving PS/PQ = TX/XR in ΔPQR with Parallel Lines

    Homework Statement At ΔPQR, a line is drawn parallel to QR, cuts PQ at S and PR at T. Line SR and QT intersect at X. Show that PS / PQ = TX / XR Homework Equations The Attempt at a Solution I can not achieve that proof. The best I can get is : PS / PQ = TX / XQ. Something wrong...
  39. M

    Electric field at the center of the equilateral triangle

    Homework Statement Consider an equilateral triangle of side 15.6 cm. A charge of +2.0 µC is placed at one vertex and charges of –4.0 µC each are placed at the other two, as shown in the diagram to the right. Determine the electric field at the centre of the triangle. I've found the...
  40. J

    Triangle and tangent line circle

    A triangle ABC, where ,<A = 60 degrees. Let O be the inscribed circle of triangle ABC, as shown in the figure. Let D, E, and F be the points at which O is tangent to the sides AB, BC and CA. And let G be the point of intersection of the line segment AE and the circle O. (but AE line not cross...
  41. A

    Triangle Math Help: Find a, b When c=4 & θ=20°

    I need help... I am taking physics for the first time and I am taking it online which probably isn't the smartest way to go...but to make matters worse I am also currently enrolled in the calculus class that is a prerequisite for the course at the same time so my math skills are not really up...
  42. K

    Calculating a Line Integral with Triangle C

    Homework Statement hello again, sorry for asking so many questions, i just want to make sure if I am correct or not calculate the line integral y^2dx+x^2dy where line C is the triangle with sides x=1, y=0 and y=x The Attempt at a Solution first of all i tried to find a customization of the...
  43. L

    Displacement & Pythagorean Theorem: Triangle ABC

    Alright, we have triangle abc with hypotenuse c. So, if you add vector a and vector b, the answer is vector c. Now, according to the pythagorean theorem, this would not make sense. But the pythagorean theorem is DISTANCE. I am guessing that this phenomenon has something to do with using...
  44. P

    Equilateral triangle coulombs law problem

    [Homework Statement 1.) Three charges are placed 25 cm apart in an equilateral triangle. Q1 has a charge of -6.0 nC, q2 has a charge of 4.0 nC and q3 has a charge of 4.0 nC. what is the force on q2? q1 is the top point, q2 is the bottom left leaving q3 on the bottom right Homework...
  45. O

    Exploring Reverse Triangle Inequality Norms

    I'm interested in thing that are norms except for the fact that they satisfy the reverse triangle inequality ||x+y|| \geq ||x|| + ||y||. The obvious example is taking p-norms for 0<p<1. Does anyone know of others or if there's any theory developed on this topic?
  46. M

    Can You Solve Euler's Triangle Formula with Just the Distance (d)?

    Hi all I have a question concerning Euler's triangle formula, where the distance (d) between the incenter and circumcenter of a triangle is given as d2 = R(R-2r), with R being the circumradius and r the inradius. I suck at algebra and I need to know how to solve this if I only know the...
  47. B

    Triangle inequality metric space

    Homework Statement Let (X,\theta) be a metric space. Take K > 0 and define. \theta : X \cross X \rightarrow \real_{0}^{+}, (x,y)\rightarrow \frac{K\phi(x,y)}{1+K\phi(x,y)} Show that (X,\theta) is a metric space. Homework Equations can someone please check my triangle...
  48. D

    Finding the Correct Parameterization for Line Integral

    Homework Statement I'm attempting Q 3 from ch 16.4 of Stewart (p 1060). We are required to find the line integral where C is the triangle with vertices (0,0), (1,0) and (1,2). The line integral is Int xy dx + x^2*y^3 dy Homework Equations The Attempt at a Solution...
  49. A

    Transforming Triangle in 3 Space with z = 1-x-y

    I am trying to transform a triangle in 3 space with the constraint that z = 1-x-y. At this point I have the following transformation matrix A, and three vector equations: | a1 a2 a3 | A = | a4 a5 a6 | | a7 a8 a9 | A*x1 = k1*y1...
  50. M

    Calculating Hypotenuse with C++ Code

    i am trying to write a code that calculates the hypotenuse of a triangle... when i try to run it, i get a run time error. I'm not exactly sure what the problem is. #include <iostream> #include <cmath> using namespace std; void hypotenuse(double side1, double side2...
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