Non-Right Triangle Solutions and Examples

High School
Geometry
Trigonometry
Non-right triangles and trigonometry

Recent questions in Non-right triangles and trigonometry

Davion Harding 2022-07-01

Given the Pythagorean triple,
$\begin{aligned} a & = \frac{x^{2} - 25}{y} \\ b & = \frac{10 x}{y} \\ c & = \frac{x^{2} + 25}{y} \end{aligned}$
so $a^{2} + b^{2} = c^{2}$ and $y \neq 0$ . We wish to set its area $G = 5$ . Thus,
$G = \frac{1}{2} a b = \frac{1}{2} \frac{10 x}{y} \frac{(x^{2} - 25)}{y} = 5$
Or simply, $\begin{matrix} (1) & x^{3} - 25 x = y^{2} \end{matrix}$
Can you help find two non-congruent right-angled triangles that have rational side lengths and area equal to 5?

Dayami Rose 2022-07-01

Find all non-right angled dissimilar triangles having integer sides and integer area simuntaneously. Are there infinitely many such triangle?

preityk7t 2022-06-27

Given a right triangle $A B C$ rotated $2 π$ , one full circle, around line $B C$ . Find area of the formed cone.
Everyone knows that the formula for a cone includes a $\frac{1}{3}$ in it due to some integral calculus.
Why can the area of a cone not be visualized as a triangle rotated about an axis? Intuitively this makes sense. The area of a triangle is $\frac{1}{2} b h$ . The distance it is rotated is $2 π$ . By that this should give us a formula $b h π$ . This seems to work for a cone with $r = 3$ and height $4$ .

abbracciopj 2022-06-26

A non-right triangle with known x,y coordinates. Lets say A,B,C is the points of triangle. I want to find a new point D which is perpendicular line from point B and lies on the line AC. Any simpled questions for D(x,y)?

Manteo2h 2022-06-26

Given a non-right triangle with vertex $C = 91^{\circ}, \bar{A C} = 39.9 c m, \bar{B C} = 32.6 c m$ , find
a) $\bar{A B}$
b) $m ∡ A$
c) $m ∡ B$
I was able to find the length of $\bar{A B}$ , which I labeled $c$
$\begin{aligned} c = & \sqrt{{32.6}^{2} + {39.9}^{2} - 2 (32.6 * 39.9) \cos (91^{\circ})} \\ c = & 51.9 c m \end{aligned}$
but I'm having a hard time finding the angle measures of $A$ and $B$ , here's my attempt:
a) $\begin{aligned} \frac{32.6}{\sin A} = & \frac{51.9}{\sin 91^{\circ}} \\ \sin A = & \frac{32.6 \sin 91^{\circ}}{51.9} \end{aligned}$
So given some theta A, we get the vertical length 0.63 radians. To find A, I used
$\begin{aligned} \arcsin (0.63) \approx 0.68 rad \end{aligned}$
So I figured I just needed to convert this to degree measure, getting me
$m ∡ A \approx {38.96}^{\circ}$
b) using the fact that the sum of angle measures of a triangle is $180^{\circ}$ , I tried to solve for angle $B$ using
$\begin{aligned} {38.96}^{\circ} + {51.9}^{\circ} + m ∡ B \approx & 180^{\circ} \\ m ∡ B \approx & {89.14}^{\circ} \end{aligned}$
but I'm getting marked wrong for both answers b) and c). What am I doing wrong?

Santino Bautista 2022-06-25

Is every triangle with 3 rational angles (in degrees) similar to one with 3 integer-length sides?

Davion Harding 2022-06-24

Let $α$ , $β$ and $γ$ be the three angles in a non-right triangle. How can i prove the following inequality?
$\frac{1 + \sin^{2} (α)}{\cos^{2} (α)} + \frac{1 + \sin^{2} (β)}{\cos^{2} (β)} + \frac{1 + \sin^{2} (γ)}{\cos^{2} (γ)} \geq \frac{1 + \sin (α) \sin (β)}{1 - \sin (α) \sin (β)} + \frac{1 + \sin (β) \sin (γ)}{1 - \sin (β) \sin (γ)} + \frac{1 + \sin (α) \sin (γ)}{1 - \sin (α) \sin (γ)}$

Bailee Short 2022-06-24

$A = (x_{1}, y_{1})$ , $B = (x_{2}, y_{2})$ , and $C = (x_{3}, y_{3})$ are three (distinct) non-collinear points in the Cartesian plane, and $a = | \bar{B C} |$ , $b = | \bar{A C} |$ , and $c = | \bar{A B} |$ . The incenter of the triangle is
$(\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c}) .$
The x-coordinate of the incenter is a "weighted average" of the x-coordinates of the vertices of the given triangle, and the y-coordinate of the incenter is the same "weighted average" of the y-coordinates of the same vertices. I am requesting an explanation for this statement.

doodverft05 2022-06-22

Given a triangle $A B C$ , $∠ C$ is $65$ degrees, side $C$ is 10. The area of the triangle is $20.$ What is the perimeter?

Bailee Short 2022-06-21

I am given 2 side lengths for one triangle and two side lengths for the parallelogram. I am asked to find the length of m (FE) and n (DE)
given the lenghts:
h (AC) = 9
k (AF) = 15
f (AB) = 16
I don't see how to use Law of Sines because I don't have any angles and I don't see how to use Law of Cosines to solve triangle ACF because I am missing a side length.

seupeljewj 2022-06-21

Consider a non-isosceles triangle, pick a vertex. Assume that the median and the altitude passing through this vertex are isogonal conjugates (i.e. symmetric w.r.t. the bisector of the angle). Prove that the triangle is right-angled in this vertex.

Davion Harding 2022-06-21

Evaluate $\csc^{- 1} (\sec 5)$ without using a calculator.

Emmy Knox 2022-06-20

Is there a non-right triangle which corresponding circumcenter is lying on one of its sides?

Boilanubjaini8f 2022-06-20

How to find length if given one side length and one angle of a non right angle triangle?

Sonia Gay 2022-06-18

Is a "triangle" with two right angles still a degenerate triangle?
A point is an example of a degenerate triangle.
In 2D, euclidean, flat space. Can the term "degenerate triangle" also describe a triangle with two right angles, an angle of 0° degrees, one side of a finite non-zero length, and two sides of infinite length?
The 0 degrees angle would be positioned on a point discontinuity and so be in two places, both infinitely far from the other two vertexes.

Extrakt04 2022-06-18

Problem: $O$ is the circumcenter of $△ A B C$ , which is not a right triangle.
$\frac{| A B \cdot C O |}{| A C \cdot B O |} = \frac{| A B \cdot B O |}{| A C \cdot C O |} = 3$ .
Find $\tan A$ . Here $A B \cdot C O$ represents the dot product of vector $\vec{A B}$ and $\vec{C O}$

xonycutieoxl1 2022-06-16

Is it possible to prove the converse of the Pythagoras theorem with out a geometric proof?.That is from $a^{2} + b^{2} = c^{2}$ to $Θ = 90^{\circ}$

Emmy Dillon 2022-06-15

If the radius of an inscribed circle in a right triangle is 3 cm and the non-hypotenuse side is 14 cm, calculate triangle's area. I've tried to use the formula: $r = \frac{a + b - c}{2}$ but that doesn't help me that much.

Finley Mckinney 2022-06-14

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From this point the surveyor moves 21.6 metres directly towards the tower and repeats the measurement. The angle of elevation of the top of the tower from this point is 68.4 degrees. Find the height of the tower.

Cory Patrick 2022-06-12

Would a right triangle with bases a=i and b=1 have hypotenuse c=0?

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.

Trigonometry

Expert Help with Non-Right Triangle Equations