All Questions
Tagged with geometry algebra-precalculus
1,320
questions
1
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258
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Sum of the vectors from centre $O$ to the polygon vertices
I'm attempting to calculate the sum of the vectors from the center of a regular polygon to each of the vertices. I have already solve it in a complex analysis manner:
To represent the vertices of a ...
- 23
2
votes
0
answers
53
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Find the vertices of a non-canonical hyperbola
Given the hyperbola
$$x^2+6xy-7y^2=20,$$ how do I find its vertices? I have found its asymptotes by factoring it into $$(x+7y)(x-y)=20$$ to obtain its asymptotes: $$y=x$$ and $$y=-\frac{1}{7}x.$$ I do ...
- 31
3
votes
3
answers
167
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How can I fit a circle into the gap between an ellipse and the coordinate axes?
Consider an ellipse with the equation $\frac{\left(x-ar\right)^{2}}{a^{2}}+\frac{\left(y-br\right)^{2}}{b^{2}}=r^{2}$. How would I fit a circle with an equation $\left(x-k\right)^{2}+\left(y-k\right)^{...
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3
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3
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243
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Inscribing two circles and an ellipse in a square
A square of given side length $S$ is to inscribe two circles and an ellipse as shown in the figure below.
If the radius of circle $(1)$ is given, determine the center and radius of circle $(2)$, then ...
- 24.2k
6
votes
3
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453
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Find the radius of the red circle
Four circles are arranged as shown in the figure below. They're numbered from $1$ to $4$.
If the diameter of circle $C_2$ is equal to $9$, and $ PT = 6 , QT = 3 \sqrt{5} $. It is also given that $UV ...
- 24.2k
0
votes
2
answers
64
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Find the length of segment $AD$ in this trapezoid [closed]
$ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. In addition, the two diagonals $AC$ and $BD$ satisfy $ AC \perp BD$. Segment $BC = 3$. And finally, the area of the blue triangle is ...
- 24.2k
1
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2
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78
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Find $x$ in this concyclic quadrilateral
$ABCD$ is a concyclic quadrilateral, with $\angle A = 60^\circ$, and $ AB = 10, BC = x , CD = x+2 , DA = x+4 $.
Find $x$.
My attempt:
Using the vector method, we can express the horizontal and ...
- 24.2k
3
votes
3
answers
182
views
Find $x$ in this quadrilateral
A quadrialteral $ABCD$ has $AB = 10$, $\angle A = 50^\circ, \angle B = 120^\circ$, $ BC = x , CD = x + 2 , AD = x + 4 $. Find $x$.
My attempt:
Applying the law of cosines to $\triangle DAC$ and $\...
- 24.2k
1
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1
answer
53
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In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$.
I was trying to solve this problem:
In an isosceles triangle with base $a$ and congruent side $b$
the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$.
After a long time of thinking ...
- 6,332
1
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1
answer
65
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Proving the implicit equation of the parabolic boundary of a plane domain defined by two parameters
$p,q \in [0,1]$ I have the equation:
$(x,y) = pq(2,1) + p(1-q)(-1,-1) + (1-p)q(-1,-1) + (1-p)(1-q)(1,2)$
I want to show that the parabolic boundary connecting $(2,1)$ and $(1,2)$ is given by
$5(x-y+1)^...
- 163
4
votes
1
answer
123
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Make $\sqrt 3$ by folding paper
I was thinking about making $\sqrt 3$ by folding (for example $A4$ paper). I do $\sqrt 2$ as a simple folding like below :
Remark: It is easy to make $\sqrt2 , \sqrt5$ by folding by $1:1$ and $2:1$ ...
- 25.2k
6
votes
4
answers
970
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Expressing the area of an isosceles triangle as a function of one of its angles.
We are given a circle with radius $1$, its center point and an inscribed isosceles triangle with $AB=AC$ and its height (as shown in the picture below). Can we express the area $(ABC)$ as a function ...
- 1,429
-1
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2
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80
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Understanding that $\sqrt{x} + a\sqrt{y} = 2$ is a branch of a parabola
Is there a simple (or simpler) way to understand that the following curve
$$\sqrt{x} + a\sqrt{y} = 2 \tag{1}$$
is a branch of a parabola?
When I say "simpler" I mean simpler with respect to ...
0
votes
1
answer
48
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practicality of an equality
Yesterday I came across this post which briefly states that $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = \frac{\sqrt{x^2 + y^2 + z^2}}{\sqrt{a^2 + b^2 + c^2}}$$ after a bit of work I managed to expand ...
- 1,429
2
votes
4
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89
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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 ...
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