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Questions tagged [plane-geometry]

Plane geometry is a subfield of Euclidean geometry, restricted to the flat two-dimensional space. Plane geometry studies shapes, ratios and relative locations of 2D figures which can be embedded into 2D plane.

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Calculate X, Y, Z of the 4 points in the 3D space.

I want to find the $x, y, z$ coordinates for $4$ points in a 3D space. Point $A$ is my origin $(X, Y, Z = 0,0,0)$ and other points $B, C, D$ are with reference to point A. I know all six distances ...
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The point of intersection of the graph of a quadratic function and a circle

Here is a question I found on the website of International Kangaroo Maths Contest. The question goes like this: A quadratic function $f(x)=x^2+px+q$ is such that its graph intersects the x and y ...
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Length and perimeter

Hello guys? I have been having constant disagreements with my fellow professors on this question. A field was to be fenced using 816 posts placed 4 meters apart,leaving a 4 meter space for the gate....
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Does the normal of any plane pass through every single point in space? If so, why?

The normal is a direction vector that is perpendicular to (edited) the plane at all points. Is the normal then able to pass through every single point in space? Edit: Suppose I can shift the normal ...
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Find angle $ \angle AED $ in the following triangle. [duplicate]

Find angle $ \angle AED $ in the following triangle. In the above triangle we have : $CA=CB ,CE=DB=BA ,\angle ACB =20^° , \angle CAB=\angle CBA=80^°$ now find $ \angle AED $. I think if we draw ...
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Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not.

Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not. Explain what this plane looks like. Would it be a plane in the xy plane? Not sure how to start this ...
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Self-Similar Polygon Tessellations

It is well-known that the only regular polygons which tessellate the plane (using only one shape) are the triangle, square, and hexagon. However, there are many more tessellations of the plane by ...
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How to study Euclidean geometry from axioms?

I want to know if there's a good book or any other type of guide to study Euclidean geometry by only the 5 axioms in plane geometry and prove every other theorems from them?
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Number of Intersection points between lower half of an ellipse and a circle

An ellipse has its axes parallel to the coordinate system axes and its major axes is parallel to X-axis. Meanwhile, there is a circle located at the coordinate system origin, whose radius is smaller ...
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Solving the equations in three variables

When I am dealing with some geometry problem in barycentric system I come across with the following equations $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}-xy\left(\frac{1}{a}+\frac{1}{b}\right)-yz\...
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Geometric proof of equivalence between two constructs of ellipse

Pretending that we don't know any analytic geometry and trigonometry. Consider the following two constructs of an ellipse, where admittedly the second one is an ad-hoc construct for the ellipse ...
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Intersection of the two planes

I need help for my vector's assignment!!! Let L be the line of intersection of the two planes x+y+z-1=0 and 2x+3y-z+2=0. Find the scalar equation of the plane that contains the line L and passes ...
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Intersection of a plane

I need help for my grade 12 Vector's homework. Can a plane be perpendicular to the x-axis and contain the line x=z, y=0? Explain. I really hope someone can answer this question
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Create an arrow shape based on a vector

I've been trying to generate an arrow shape with javascript, which would have following characteristics: random direction random length (in bounds which doesn't matter in this question) the length of ...
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Which matrices perform dilation?

For each angle $\theta \in \mathbb{R}$, we get two corresponding matrices: $$\mathrm{Rot}^\theta = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos\theta\end{bmatrix}, \qquad \...
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In $\bigtriangleup OAB,\angle\mathrm{AOB}=90^\circ.$ Let C lie on segment $AB$ such that $\overrightarrow{OC}\;\perp\;\overrightarrow{AB}$.

I am stuck at proving the following question. In $\bigtriangleup OAB,\angle\mathrm{AOB}=90^\circ.$ Let C be the point on the segment $AB$ such that $\overrightarrow{OC}\;\perp\;\overrightarrow{AB}$. ...
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Given a plane and a line, find the equation of another plane that has an angle 30 of degree to the given plane and contains the given line.

Here's what I did: For case 1, 2 and 3, I'm ok now, but for case 4, I still have trouble finding the right way to solve it. My solution to case 4 is theoretically possible, but as for me, I've no idea ...
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Distance of a plane

I have posted this question in Stack Overflow programming forum. Someone there feels it might be more suited to Mathematics. I have to warn you I am rusty in math and was terrible in Algebra. ...
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Make a formal description of a volume

I'm attempting to write formally the description of a volume. For an academical definition for engineers. Can you give me your feedback if it is correctly defined? In case you find it relevant, the ...
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How to show the product of two reflections is a translation?

If $\ l = P+ [v]$ and $\ m = Q+ [v]$ are lines. And $\ |v|=1 $, then $$\Omega_l\Omega_m= \tau_w$$ where $\ w= 2<P-Q,V^\perp>V^\perp $ and also $$\Omega_m\Omega_l=\tau_{-w}$$ My work: If $\ l$ ...
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Line and distance equation

I've studied line geometry especially one that has to do with distance formula and question. But I just don't know how to approach this question Find the equation for the set of all points ...
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Approach on solving a word problem based on cyclic geometry

Here is the question - Two chords AB and CD of a circle with centre O , Intersect each other at P. If angle AOD = 100 and angle BOC= 70. Find value of angle APC. Now I know how to solve this kind of ...
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When a third vector in a plane does not lie in the span of 2 linearly independent vectors in the plane

For instance, can the 3 vectors $\vec a=[1, \ 0, \ 1]^T, \vec b=[2, \ 7, \ -2]^T, \vec c=[3, \ 1,\ 5]^T$ lie on the same plane in $\mathbb R^3$? My understanding is that the span of 2 linearly ...
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Prove that $n$ lines form $[n(n+1)/2]+1$

I saw the same formula on the web as $(n^2+n+2)/2$. I saw the same problem here. However, I didn't understand very well. I have to prove it and it seems I should use induction. So I decided to do it "...
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distance between intersection of two planes and origin

I have a following problem. I want to find the smallest possible distance between the line intersection of two planes given by: \begin{equation} x + 2y−2z = 3 \text{ and } 2x + y + 2z = 6 \end{...
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How to solve the following differential equation of a figure of eight?

The following differential equation describes the first derivative of an equation that describes an object moving in the X-Y plane along figure of eight with constant speed: $ 4(x-h)^3 \frac{dx}{dt} ...
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What is the maximum number of regions in 3D space a plane can intersect?

Since there are 8 regions or"quadrants" I thought it would be 6 regions as the max. I do not know if I am right.
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What are some nice proofs-by-cutting-and-pasting?

I enjoy geometric proofs made by cutting-and-pasting. There are some famous examples for Pythagoras's Theorem. Here's another example for the Law of Cosines. Do you know some other nice proofs using ...
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Equation of Plane involving Intersection of planes

Find a plane through $A$ $(2, 1, -1)$ and perpendicular to the line of intersection of the planes $2x + y - z = 3$ and $x + 2y + z = 2$. Not too sure of what to do here, I know I need to find the ...
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Do these lines in 3D space intersect?

Th lines formed by $(0,0,0)+ \lambda(1,1,1)$ and $(0,6,0)+ \lambda(0,-3,2)$ ever intersect? It seems like the do but they don't. How do I show this algebraically?
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Intersection of two open connected sets in the plane

Assume $O_1$ and $O_2$ are two open connected sets of $\mathbb{C}$ such that $O_1\cup O_2=\mathbb{C}$. Is it true that $O_1\cap O_2$ is connected ?
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What are the intercepts of the planes $x = 0$ and $2y + 3z = 12$?

What are the intercepts of the planes $x = 0$ and $2y + 3z = 12$? The word intercept is confusing me because I don't understand if I should say they intersect at point $(0,6,0)$ or the intercept is at ...
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Equations in Three Dimensions

Graph the planes x – y = 0 and y – z = 0. Then graph the line "a", if it exists, defined by the intersection of x – y = 0 and y – z = 0. Describe line "a" including all its intercepts, if it exists. ...
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How can I prove the betweenness of the tree straight segments

I come across an elementary geometry problem as following: Assume that $\angle CAD=12°, \angle CBD=24°, \angle CAB=36°, \angle ABD=48°.$ Estimate that $\angle ACD=?$ And I tried so far: Construct ...
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How to map a point in plane A to a point in plane B

I have a 3D point $P$ from Microsoft Kinect (relative to the Kinect camera plane - call it plane $A$), and I want to find point $P'$ that's mapped to $P$ on another plane (call it $B$) that I have an ...
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Circumradius of a regular heptagon

My graphing software says that the value of circumradius of a regular heptagon, of side unity, upto 5 decimal places, is 1.15238. Just as the circumradius of a regular pentagon of length unity can be ...
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Meaning of $a^2 + b^2 + c^2 = 1$ in a normal form plane equation

As stated above. I know that the $a, b, c$ represent the normal vector of the plane and that you can normalize them so that $a^2 + b^2 + c^2 = 1$. But what is the main reason for doing the ...
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noncoplanar points define 3-space

It takes four noncoplanar points to define a 3-space. Explain whether each of the following is true or not and explain why: a) Two skew lines define a 3-space. True because skew lines are Two or ...
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Given the angle between planes $\pi_1$ and $\pi_2$ is equal to the angle between…

I'm not sure where I'm going wrong with this question but i keep coming to a hexic equation rather than a quartic equation. the three planes: $$\pi_1: ax+2y+z=3$$ $$\pi_2: x+ay+z=4$$ $$\pi_3: x+y+...
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parallel lines and a plane

Explain why two parallel lines define a plane. If I hold two pencils so that they’re parallel, there’s only one position in which a plane can rest on both pencils.But can someone give me a more valid ...
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A line in the xy-plane contains the points (5, 4) and (2, –1)

Question: "A line in the $xy$-plane contains the points $(5, 4)$ and $(2, –1)$. Which is bigger: a) the slope of the line or b) $0$." Result: They draw out the figure and say "you can see that the ...
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Equation of Plane through Center

I have a pentagon for which 3 vertices were chosen to compute the equation of the plane. How to I find the normal passing through its center? $$P_1 = [ 3.096, \ 0.492, \ 3.287]$$ $$P_2 = [ 3.118, ...
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Single Point in other Dimensions

Would a single point and a fixed distance determine a unique segment in 2-space or 3-space like it does in 1-space when given the length of the segment and location of its midpoint? Please explain ...
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equation of a plane with 3 points on axis

Well I'am stuck with one question about finding equation of a plane.. Few months ago I got at my exam question that sounded something like this: Find the equation of a plane if it cuts segment on x-...
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How to get equal distance planes from a plane

I have 3 parallel planes and the normal vector 'W'(A,B,C) and this is normal to all the 3 planes. 3D planes Top plane equation is Ax+By+Cz+D=k Middle plane equation is Ax+By+Cz+D=0 Below plane ...
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Find the maximum percentage of acute angled triangles in a plane with $100$ points ; no $3$ of which are collinear.

In a plane there are $100$ points, no three of which are collinear. Consider all possible triangles having these points as vertices. Find the maximum percentage of these triangles which are acute-...
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What is the name of the following theorem?

I chanced upon this interesting theorem which states the following: On the sides of parallelogram $ABCD$, squares are constructed exterior to it. Then, their centers, $M_1, M_2, M_3 \text{ and } M_4 $...
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circumcenter coincides with center of mass

I am trying to prove the following statement. Any suggestions or references are highly appreciated. Consider $n$ points in $R^2$, i.e., $x_i\in R^2, i=1,\ldots, n.$ Suppose the centroid (or center ...
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I am not understanding this math about alternate angles

This is the math. This is the given solution :- This and this Now, my problem here is, $\angle ABC$ and $\angle DAC$ are supposed to be alternate angles here, and thus equal. But as far as I know,...
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Cartesian Derivation of Cartesian Equation of 3D Plane

The highschool Math textbooks that I have seen, derive the equation of a 2D line y = mx + c or, equivalently, ...