What is Right triangle: Definition and 88 Discussions

A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle (Greek: ὀρθόςγωνία, lit. 'upright angle'), is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of the right angled is the basis for trigonometry.
The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B.
If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

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1. Find the sides of the right triangle so that their sum is minimized

Hello. We know that ##a^2+b^2=c^2## and we want to minimize ##a+b##. $$b= \sqrt {c^2-a^2}$$ $$\dfrac {d}{da} (a+\sqrt {c^2-a^2})=0$$ $$1-\dfrac {a}{ \sqrt {c^2-a^2}}=0$$ This gives $$a=\dfrac {c}{\sqrt 2}$$ But it doesn't work for c=5. I know a=3 and b=4 minimize a+b.
2. I Finding the center of area (centroid) of a right triangle

To find the y value of the centroid of a right triangle we do $$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$ What is wrong with using $$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
3. MHB How to Find the Length of the Hypotenuse of a Right Triangle?

AB=3, DC=5, ∠ CAD=$45^o$, AB ⊥ BC. Find the length of AC.
4. I Properties of the legs of a right triangle

I want to know if a right triangle can only have one leg that is a perfect power of a number. Another words is it impossible for a right triangle to have two legs that are numbers that are raised to the same perfect power? Can somebody answer this question and show me the proof?
5. Perpendicular Bisector of a triangle

Here is my attempt to draw a diagram for this problem: I'm confused about the "the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##" part of the problem. How does perpendicular bisector of ##BC## cut the side ##CA##?
6. What is the angle needed to solve this right triangle?

The Figure My Attempt at Solution ##\tan{ACB} = \frac{AB}{BC}, \ \tan41.45^\circ = \frac{AB}{10} \Rightarrow AB = 10\tan45.41^\circ \approx 8.83##cm Similarly ##\tan{CBD} = \frac{CD}{BC}, \ \tan32.73^\circ = \frac{CD}{10} \Rightarrow CD = 10\tan32.73^\circ \approx 6.43##cm After this I...
7. To find the ratio of forces around a right triangle

Could I please ask for help with the following: ABC is a right-angled triangle in which AB = 4a; BC = 3a. Forces of magnitudes P, Q and R act along the directed sides AB, BC and CA respectively. Find the ratios P:Q:R if their resultant is a couple. Book answer is 4 : 3 : 5 Here's my diagram...
8. MHB Solve Right Triangle Problem Without Knowing Bottom Line

How would I have to calculate this question for an answer, a friend of mine told me he could get the answer without knowing that the bottom line was 16 meters, I can't seem to find a way that would work, I am not sure if I am missing something or he is lying.
9. MHB Finding the Leg Lengths of a Right Triangle with an Acute Angle of 22°

A right triangle has an acute angle measure of 22°. Which two numbers could represent the lengths of the legs of this triangle? OPTIONS a. 2 and 5 b. 1 and 5 c. 3 and 5 d. 4 and 5 I know that each leg represents the sides of the right triangle opposite the hypotenuse. I think the tangent...
10. Project Motion/Trigonometry Question

My reasoning and answer is wrong, but I cannot figure out why. Perhaps it is strange, perhaps not, but I want to figure out why my initial method of solving this problem did yield an incorrect answer. I began by creating an equation and drawing a right triangle. x is the horizontal part of...
11. Find the electric field at the point P on a right triangle

Homework Statement https://www.physicsforums.com/threads/find-the-electric-field-in-the-point-p-of-a-right-triangle.965285/#post-6125768 knowing that the three charges are equal and that the angles of the triangle are 90°, 45°, 45°. Homework Equations The Attempt at a Solution I tried...
12. MHB Find C's Coordinates for Right Triangle ABC

Right triangle ABC. BC = a = sqrt(17) AC = b = sqrt(68) AB = c = sqrt(85) A's coordinates: 0,12 B's coordinates: 6,5 What's EASIEST way to get C's coordinates?
13. MHB Length of sides of right triangle with 1 angle and 1 side

Hello so I'm a high school student and I came up with this question and I wanted to know if this was possible to do? So I tried to research and find a way to find the length of DC and I couldn't find anything, so I am here to ask for help, is this possible? I figured it would go in the...
14. MHB Integer-Sided Right Triangle: 2001 Leg Length & Minimum Other Leg Length

The shorter leg of an integer-sided right triangle has length 2001. How short can the other leg be?
15. MHB Is the perimeter of a right triangle equal to its area? (Part 2)

A right triangle is given. One leg is u units and the other leg is v units. The hypotenuse is given to be w units. If u = [2(m + n)]/n, v = 4m/(m - n), and w = [2(m^2 + n^2)/(m - n)n, show that (1/2)(uv) = u + v + w I must multiply u times v times (1/2), right? I then must add u + v + w. The...
16. MHB Can Pythagoras' Theorem Prove this Right Triangle Hypotenuse Equation?

A right triangle is given. One leg is u units and the other leg is v units. The hypotenuse is given to be w units. If u = [2(m + n)]/n and v = 4m/(m - n), show that w = [2(m^2 + n^2)/(m - n)n. Must I square u and v to show that w = [2(m^2 + n^2)/(m - n)n?
17. MHB What is the multiplier for finding the area of triangle ADG?

Not sure how to use a single area and line segments that are same to calculate the areas and line segments for the areas.
18. MHB Draw Sketch: Right Triangle Area = Half Parallelogram Area (Party)

Any thoughts on the sketch? (Party) Many Thanks :)
19. Build right triangle with two points and a line

Homework Statement Given the points ##A (1, -1, 0)## and ##B (4, 0, 6)##, find the point ##P## of the line ##s## so that the triangle ##ABP## is a right triangle in ##B##. Calculate the area of the triangle. ##s : \begin{cases} x = 1 + 4t \\ y = 2 - 3t \\ z = 3 \end{cases}## ##\vec v_s = (4...
20. Prove this is a right triangle in a sphere

Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
21. MHB Right triangle, feet of altitude, angle bisector and median

Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$ My solution. Can you check it is...
22. B Proofs: Hypotenuse is the longest side of a right triangle

I want to prove that the hypotenuse is the longest side of a right angled triangle. Could people check that the proof I'm giving is correct? Say the hypotenuse is of length ##c## and the other two sides are of length ##a## and ##b##. First of all, we obviously have: ##a^2 + b^2 > a^2 \quad##...
23. MHB Can you solve this right triangle puzzle with a unique angle ratio?

B b-a D a c-b+a C b A Right triangle ABC, with the standard a, b, c side lengths. Angle BAC = 2u degrees. Point D is on the hypotenuse AB, such that: BD = b-a, angle BCD = 3u degrees. Calculate u.
24. I Right Triangle and the Pythagorean Triple Formula

Given: A right triangle and all the sides of the triangle are whole numbers. Does this imply that all the sides of the triangle can only be found by using the Pythagorean triple formula? Another words, is it possible that a right triangle can exist with whole number sides that escape the...
25. MHB Right Triangle: Proving It's a Right Triangle

The sides $x,\,y,\,z$ of a triangle satisfy the equality $(x^4+y^4+z^4)^2=2(x^3+y^3+z^3)$. Prove that it's a right triangle.
26. Getting a triangle from trigonometric function

How can I get a right triangle from the inputs and outputs of trigonometric functions? For example: sin(x) = y The triangle would have one angle as x and the opposite edge of the triangle would be y/hyp etc. How can I get all of these values from any trigonometric function? Please tell me if I...
27. Calculate the Unknown Angle of a Right Triangle

The problem A right triangle has an angle a and we know that ##cos \ a = \frac{1}{3}##. What is ## tan \ (90°-a) ## The attempt I know that the ration between the adjacent side and the hypothenuse is 1/3. I am not interested in the real lengths of the sides. I can therefore calculate the...
28. Magnetic Force of a Right Triangle influenced by a Line

Homework Statement http://www.sumoware.com/images/temp/xzlknterambqmokp.png How to calculate the Magnetic Force of THE "Right Triangle" influenced by a Line when the magnetic field isn't constant B=u0*I/2piR? Homework Equations B=u0*I/2piR F=iLB The Attempt at a Solution I can use F=iLB to...
29. Solving Laplace's Equation for 2D isosceles right triangle

Homework Statement Find the two-dimensional solution to Laplace's equation inside an isosceles right triangle. The boundary conditions are as is shown in the picture: The length of the bottom and left side of the triangle are both L. Homework Equations Vxx+Vyy=0 V=X(x)Y(y) From the image...
30. 2 Cubic Blocks on a Right Triangle

Homework Statement A right-triangular wooden block of mass M is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass m, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger...
31. MHB Distance Between Intersection Points of Bisectors & Medians in Right Triangle

The legs of chateti of a right triangle are 9 and 12 cm. Find the distance between the intersection point of bisectors and the point of intersection of the medians
32. The area of a square with right triangle inside it

Homework Statement I need to find the area of the square in the following figure: Homework Equations Basic Trig relations. The Attempt at a Solution I aimed to find the length of BC, but first I had to find the unknowns of the right triangle CDE, which are EC=5m, <DCE=36.86ْ , <DEC=53.13ْ ...
33. Finding the Area of a Similar Right Triangle

For example: if it was given that two right triangles are similar triangles and that the hypotenuse of one is twice as long than the other how would you find the area of the triangle with the twice as long hypotenuse given the area of the other? Similar right triangles means they are the same...
34. Where is the right triangle in sides a, b, c, for a hyperbola?

An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values...
35. Area of a right triangle with little data

Homework Statement Homework Equations x^2 + y^2 = 9 A = 0.5xy x ≠ y The Attempt at a Solution x^2 + y^2 = 9 A = xy/2 (x + y)^2 = x^2 + 2xy + y^2 = 9 + 2xy = 9 + 4A A = ((x+y)^2 - 9)/4 Then I am lost. I need to find the area.
36. Why can't we apply trig ratios to any angle of a right triangle?

Why do trigonometric ratios have to be related to the angle between the base and hypotenuse of a right angle triangle? I am trying to understand why I can't use these ratios to any angle of a right angle triangle. I try to do that in the attached document. It seems to work for all ratios...
37. MHB Maximizing the volume of a cone formed by revolving a right triangle

A triangle hypotenuse given rectangle is rotated around one of their legs to generate a right circular cone? find the cone of greater volume. resp V= (2Sqrt(3)pi L^3)/27 It says hypotenuse given but it has no value According to the answer you can name it L
38. Optimization - rectangle inscribed in a right triangle

Homework Statement A rectangle is to be inscribed in a right triangle having sides 3 cm, 4 cm and 5 cm, as shown on the diagram. Find the dimensions of the rectangle with greatest possible area. Homework Equations 1. x^{2}+y^{2}=w^{2} in terms of w=\sqrt{x^{2}+y^{2}} 2...
39. MHB Prove Triangle Inscribed in Semicircle Is Right Angle | Arundev Answers

Here is the question: I have posted a link there to this thread so the OP can view my work.
40. How to determine angle of net force-electric charges of right triangle

Homework Statement The .274 and .88 was found using the equation of electrostatic force . Homework Equations K=q1q2/d^2 The Attempt at a Solution Would I just tan inverse of .274N and .88N which would be 17 degrees. However my question is: how to determine the reference pt: would...
41. Use vectors to form a right triangle on a circle

Homework Statement Use vectors to demonstrate that on a circle any two diametrically opposed points along with an arbitrary third point(on the circle) form a right triangle Homework Equations Hint: assume without a loss of generality that the circle is centered at the origin and let v...
42. Trig Right triangle Trigonometry

Homework Statement As ship is anchor off a long straight shoreline that runs north and south. From twi observation points. 15 miles apart on shore the bearings of the ship are N 31 ° E and S 53 ° E. What is the shortest distance from the ship to the shore. Homework Equations Sin θ Opp/...
43. MHB Simplify a trig. expression using a right triangle

I have the expression sin^{-1}(cosx) I'm not sure how to simplify this at all. I've never done a problem like this and it's in my textbook as a review question. A quick boot in the right direction would help
44. Did you know that a right triangle can be found using only one side?

I have found a way to find right triangles using only one side. How do I show you without being deleted?
45. Is Sin^2(a) Equal to Sin^2(b) + Sin^2(c) Only in a Right Triangle at A?

Homework Statement prove that sin^2(a)=sin^2(b)+sin^2(c) if and only if ABC is a right triangle in A i worked really hard on this one I'm really confused why i didn' get the answer Homework Equations The Attempt at a Solution a+b+c=pi tried turning everythng to cos 2x didn't helpi...
46. Finding Volume of Solid: Isosceles Right Triangle Cross-Sections

Hi, I'm still practicing how to find volume. 1. My problem is this: "Find the volume of the solid described below: The base of the solid is the disk x^2 + y^2 ≤ 4. The cross-sections by planes perpendicular to the y-axis between y=-2 and y=2 are isosceles right triangles with one leg in the...
47. Relative Motion problem that doesn't have right triangle relationship

Homework Statement a) Find the speed of the boat with respect to the Earth. (km/h) b) Find the speed of the boat with respect to the river if the boat's heading in the water is 60° south east (km/h) Homework Equations v(x) = v cos(θ) v(y) = v sin(θ) v = √(vx^2+vy^2) but can't use it in...
48. In a Right Triangle you have one angle - how do you find the other?

If you have a Right Triangle and you have one angle, alpha. How do you find the other angle, beta?
49. Right triangle with complex vertices

Homework Statement Find the geometric image of the complex number z, if z, z^2, z^3 are the vertices of a right triangle. Homework Equations The Attempt at a Solution I tried expanding z^2, z^3, and than using both the pythagoras theorem, and vectors (in separate attempts), but...
50. Center of mass of right triangle

I've been reading that the center of mass of a right triangle - the coordinates of the COM, is (1/3b,1/3h)- I can't for the life of me figure out why this is. Is there some sort of clear proof I can take a look at? I don't really know what to integrate..