Non-Right Triangle Solutions and Examples

High School
Geometry
Trigonometry
Non-right triangles and trigonometry

Recent questions in Non-right triangles and trigonometry

tapiat0wa4c 2022-06-04

By doing some right triangle gymnastics, we can derive things like
$\cos (\arctan x) = \frac{1}{\sqrt{1 + x^{2}}}$ , for $x > 0$ $\cos (\arcsin x) = \sqrt{1 - x^{2}}$
$\cos (\arcsin x) = \sqrt{1 - x^{2}}$
$\tan (\arcsin x) = \frac{x}{\sqrt{1 - x^{2}}}$
What about $\arctan \cos (x)$ , $\arcsin (\tan x)$ , etc?

Schenone2pare 2022-06-01

What are some non-trivial ways to construct a similar triangle from a right triangle?
By non-trivial, I mean not:
- The triangle itself or any translations or rotations of it
- A 'shrinking' of the right triangle; that is, for $△ A B C$ with $∠ A B C = 90^{\circ}$ , given points $D$ on $\bar{A C}$ and $E$ on $\bar{B C}$ such that $∠ D E C = 90^{\circ}$ we have $△ D E C \sim △ A B C$ , along with any arbitrary renaming of points. Or, more generally, any scaling of the triangle.
One of the most well-known ways to construct a similar triangle is by dropping a cevian from the hypotenuse to the vertex which is perpendicular to the hypotenuse. In other words, an altitude.
I can only think of one more case, and I'm not sure if it is true (verification would be appreciated); given right triangle $△ A B C$ , let $D$ be the midpoint of $\bar{A C}$ and $E$ be on $\bar{B C}$ such that $∠ B E D = 90^{\circ}$ . Then, $△ B E D \sim △ A B C$ . In fact, I think this triangle is congruent to the one mentioned in the last paragraph. Also not sure on that, though.
The last one came about as a surprise while I was working on a proof, but I could not prove it nor could I think of any other ways to construct a similar right triangle to one given. So, my question: given a right triangle, what are some ways to construct a similar triangle (preferably ones that you found clever/interesting)?

ht1o4qgqdy 2022-06-01

How do you prove a triangle with the hypotenuse of length 5 and other sides with lengths 3 and 4 is a right triangle?

Brooke Kramer 2022-05-28

In the triangle $A B C$ , $a = 5$ , $b = 6$ and the area is $11 {cm}^{2}$ . Find the size of Angle $C$

Scolfaro2y 2022-05-26

Expressing the arctan function in a different form?
$\arctan (\frac{1}{ω}) = \frac{π}{2} - \arctan (ω)$

Trevor Wood 2022-05-25

Can we find two non-congruent right triangles with whole-number lengths and congruent hypotenuses?

Briana Petty 2022-05-24

Given: $△ A K M, K D ⊥ A M$ , $△ A K M, K D ⊥ A M$ Find: $K D$

Isaiah Farrell 2022-05-22

Prove the following trigonometry identity
For any $x \in [0, 1]$ show that
$\arcsin (x) + \arccos (x) = \frac{π}{2}$

lurtzslikgtgjd 2022-05-13

Which area is more fundamental: parallelogram or triangle?

poklanima5lqp3 2022-05-09

I have a triangle that I know the lengths of all the sides. But I need to know the angles. Needs to work with non-right triangles as well.
I know it is possible, and I could have easily done this years ago when I was in trig, but it has completely slipped my mind.
I'd prefer a solution that I can code into a function, or something that does not require constructing right triangles from it.

lurtzslikgtgjd 2022-05-08

Let $A$ be the set of all $d$ such that $d < 1$ and $d = \frac{a}{b}$ where $a^{2} + b^{2} = c^{2}$ and $a, b, c \in I$ . Is A an infinite or finite set and how can that be proven?

Dominick Blanchard 2022-05-07

The base of a pyramid is a right triangle with the longest side = 12cm. All non base edges are also equal to 12cm. And how can I find the height of the pyramid?

The area A of a triangle is given by bab sin 0, where a and b are the lengths of two sides and O is the angle between these sides. Suppose that a = 5, b = 10 and 0 = . (a) Find the rate at which A changes with respect to a if b and 0 are held constant. (b) Find the rate at which A changes with respect to 0 if a and b are held constant. (c) Find the rate at which b changes with respect to a if A and O are held constant.

Zaccagliaodi 2022-03-02

The triangle ABC is an isosceles triangle. What are each of the congruent base angles if the vertex angle measures 40 degrees?

Brooklyn1wp 2022-02-28

If the hypotenuse of a triangle is 12 cm and the measure of its legs are in the ratio $1 : 2$ , then what are the measures of it’s legs?

Shayla Lyons 2022-02-14

The length of the base of an isosceles triangle is x. The length of a leg is 4x- 2. The perimeter of the triangle is 104. Find x.

Shayla Lyons 2022-02-05

Write about TWO important mathematical books/publications (give author, title, short content description) from EACH of the following 2time periods so a total of 4 publications/books.
1300-1750; and 1750-2000.

Ishaan Mcneil 2022-02-04

One of the largest issues in ancient mathematics was accuracy — nobody had calculators that went out ten decimal places, and accuracy generally got worse as the numbers got larger. The famous Eratosthenes experiment, that can be found at famousscientists,org/eratosthenes/, relied on the fact known to Thales and others that a beam of parallels cut by a transverse straight line determines an equal measure for the corresponding angles. Given two similar triangles, one with small measurements that can be accurately determined, and the other with large measurements, but at least one is known with accuracy, can the other two measurements be deduced? Explain and give an example.
The similarity of triangles gives rise to trigonometry.

How could we understand that the right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles? Ultimately, the similarity of triangles is the basis for proportions between sides of two triangles, and these proportions allow for the calculations of which we are speaking here. The similarity of triangles is the foundation of trigonometry.

Adrienne Gross 2022-02-04

Triangle ADEF is a Right triangle, where angle F = 90, side d = 19, side f = 38. Find the measure of angle D. Use knowledge of triangles and/or trigonometry to justify your answer.

Elaina Conner 2022-02-03

In this section you are to provide short answers to each question. 1. (a) Identify each as a line, line segment or ray. (i) AB (ii) -→AB (iii) ←→AB (b) give reasons for your answers in 1a (c) Are the -→AB and -→BA same? Why? (d) Does a regular pentagon tessellate? ........ (e) Justify your answer in 1d Fill in the spaces in 1f to 1h (f) A ....... divides a two-dimensional shape into two congruent shapes. (g) When two straight lines cross, it is found that the angles opposite each other are the same size. They are known as ............. (h) Can the sine rule relationship in trigonometry be used with nonright-angled triangles? ......

Before you complain that Trigonometry is not something that you would like to do, it is good to know that non-right triangle equations, among other objects, are constantly used in video games, robotics, and engineering where scientists are looking for answers to their questions and apply various equations to receive the best results for symmetric building and other tasks. Take your time to study equations for non-right triangles based on examples and seek help by learning through our solutions that are addressing the most popular questions and problems where non-right triangles are implemented.

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