All Questions
Tagged with triangles algebra-precalculus
166
questions
1
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46
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transforming a polynomial function
I was exploring transforming polynomials (sorry if this is the wrong term). Essentially, I found a way to rewrite polynomials in different equivalent forms analogous to changing a quadratic from ...
- 53
2
votes
4
answers
89
views
If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$
A circle of radius $1$ unit is inscribed in $\triangle ABC$. If $BC=3$ then find the least possible area of $\triangle ABC$ and also find the perimeter of the triangle when it has the least possible ...
- 9,599
0
votes
1
answer
41
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Connection between trigonometric ratios and similar triangles.
Working on:
Daniel J. Velleman. (2024). "Calculus: A Rigorous First Course" (p. 63)
The author explains:
Although we have not used right triangles to define the trigonometric functions, ...
- 2,367
4
votes
1
answer
152
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Does every triangle satisfy $\frac{1}{a-b+\pi R} + \frac{1}{b-c+\pi R} - \frac{1}{c-a+\pi R} < \frac{1}{R}$?
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,
$$
\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{R}{a-b+Rx} + \frac{R}{b-c+Rx} - \frac{R}{c-a+Rx}...
- 20.8k
5
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2
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158
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Does every triangle satisfy $a^c + b^c - c^c < \pi$
Let $(a,b,c)$ be the sides of a triangle and let its circumradius be $1$. Is it true that
$$
a^c + b^c - c^c < \pi
$$
My progress: In the special case where at least two of the three sides are ...
- 20.8k
-1
votes
2
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103
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How to find the side lengths of a triangle in the incircle of a Pythagorean triple
Given a Pythagorean triple with an incircle, how do I find the sides of a triangle connecting the triple's tangents to that incircle. In the diagram below, I know how to find side $\,i\,$ but not the ...
- 6,416
0
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1
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47
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Does this relationship proves that the given positive numbers are equal?
Suppose I have six positive numbers divided into two sets given by: $\{a, b, c\}$ and $\{d, e, f\}$. Now, I have relations as:
$$a^2-b^2=d^2-e^2$$
$$b^2-c^2=e^2-f^2$$
$$a^2-c^2=d^2-f^2$$
I some feel ...
- 587
2
votes
0
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71
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How to algebraically prove the sum of the projections of two sides of a triangle is equal to the projection of the third side of the triangle?
Look at the picture above, graphically/geometrically, the green line is equal to the sum of the blue and red lines. But I am trying to think of an algebraical proof. Let's say the lengths of the side ...
- 225
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0
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53
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Vertex circle radii of Pythagorean triples
@Blue was kind with his comments on a previous question here.
I'd now like to share some new relationships I found using algegra and my favorite formula for generating Pythagorean triples. In this ...
- 6,416
18
votes
7
answers
2k
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Showing that the centers of two semicircles and a circle inscribed in a quarter circle form a right triangle
The challenge in this image is to determine the radii of the two semicircles and the full circle.
Determining the radii of the two semicircles is straightforward; if the radius of the quarter circle ...
- 189
1
vote
1
answer
88
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Smallest possible value of $k$ such that the roots of $x^2-127x+k=0$ are positive integers [closed]
In a triangle, two sides have equal lengths both shorter than the third side. The length of the three sides are all integers and all satisfy the equation $x^2-127x+k=0$, $k$ is a constant. Find the ...
- 13
3
votes
1
answer
210
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Finding the height of the building using Trigonometric Ratio.
Question:
At a certain point in a large, level park, the angle of elevation to the top of an office building is ${30}^{\circ}$.
If you move ${400}ft$ closer to the building, the angle of elevation is $...
- 191
7
votes
3
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221
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In $\triangle ABC$, prove: $\frac{\sin^2 \frac A2}{\sin B \sin C}+\frac{\sin^2 \frac B2}{\sin A\sin C}+\frac{\sin^2 \frac C2}{\sin A\sin B} \ge 1$
In a $\triangle ABC $, prove : $$\frac{\sin^2 \dfrac{A}{2}}{\sin B \sin C} + \frac{\sin^2 \dfrac{B}{2}}{\sin A\sin C} +\frac{\sin^2 \dfrac{C}{2}}{\sin A\sin B} \geq 1 $$
My approach :
$$2\left(\frac{\...
- 117
2
votes
0
answers
50
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Radii of touching spheres centered on Pythagorean triple vertices.
There are similar questions and answers out there but other answers I have seen seem unnecessarily complicated.
I came up with what I think is a simple approach some time ago but I'm now doubting my ...
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-1
votes
1
answer
136
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Solution Check for a Related Rates Problem Involving Shadow Length and Position
I am working on the following problem:
A girl 5 ft tall is running at the rate of 12 ft/s and passes under a
street light 20 ft above the ground. Find how rapidly the tip of her
shadow is moving when ...
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