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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 in a 2D plane.

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What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?

For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let $$ \lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }. $$ And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
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Calculate the angle $x$ in the quadrilateral $ABCD$

Calculate the angle $x$ formed by the diagonals of a quadrilateral given $$\angle ABD = 50^o\\\angle EBC = 80^o \\ \angle BDA = 100^o \\ \angle BDC=30^o\\\angle CEB = x$$ (Answer:$60^o $) My try: $\...
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Calculate the angle $x^o $ in the triangle below

Calculate the angle $x^o $(Answer:$25^o$) I try: $angle ABE = 180^o -65^o = 115^o $\ $\angle EBC = 180^o - 70^o - 45^o = 65^o \implies \angle ABE = 50^o$ $ \angle AEB = 180^o - 70^o =110^o $ $\angle ...
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Finf the segment EC in the triangle ABC below

In a triangle $ABC$ where $AB = 6$ and $BC = 9$; the extension of the bisector of angle ABC intersects the (perpendicular) bisector of $AC$ at $P$ and is then drawn: PE⊥BC. Calculate $EC$. (Answer:1,5)...
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Is there any two-coordinate system where X, Y is the same point as Y, X? [closed]

Is there any two-coordinate system where e.g., (10,36) is the same point as (36,10)? You see we here at the Rescue Centre are at our wits' end. Half our reports come in with latitude first, half with ...
Dan Jacobson's user avatar
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Find the measure of $\measuredangle BAC$ in the triangle below given two sides and an angle

Em um triângulo $ABC$, se $∡𝐴⁢𝐶⁢𝐵=34^𝑜⁢30′$; $AB = 6$ e $BC = 10$. Calcular ∡𝐵⁢𝐴⁢𝐶. (Answer:$71^o 30'$) I try: T.coss: $6^2 = 10^2+AC^2-2.6.AC.cos \angle C \implies 12AC cos \angle C C$ $\...
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How to prove opposite angle bisector theorem for convex quadrilaterals?

Let $ABCD$ be a convex quadrilateral with $BL$ and $DL$ be its angle bisectors. I want to know how to prove that the acute angle $\alpha$ between these bisectors is equal to $\frac{\left|\angle A - \...
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circle passing over another circle

Imagine a circular laser beam of light of diameter d1 moving at velocity v passing over a stationary circular opening of diameter d2 along the same plane holding both circles. The two circles are ...
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Prove centre is inside cyclic quadrilateral with perpendicular diagonals

Let $ABCD$ be a convex cyclic quadrilateral, and the diagonals $AC$ and $BD$ are perpendicular. The circumcircle of $ABCD$ has centre $O$. I am trying to prove that the centre $O$ is inside $ABCD$. I ...
wenbang's user avatar
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{SELF STUDY HOMEWORK] [IM 3] Need a hint for rewriting the ellipse equation as the circle equation.

I will copy and paste the problem in quotations. "Since a circle is a type of ellipse it stands to reason that the ellipse equation can be rewritten into the circle equation. If this is possible, ...
limaosprey's user avatar
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Determine the distance from a point $P$ to the center of the circle as a function of the radius $R$

Let AB be the diameter of a circle with center O and radius R. On the extension of AB we choose a point P (PB<PA). Starting from P we take a secant that cuts the circle at points M and N (PM<PN),...
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Find the side size of the square in the figure below

In the plane figure below, we have a parallelogram $ABCD$ and three squares, two of which have side measurements equal to the sides of the parallelogram and the largest of which has side measurements ...
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Sierpinski Gasket coordinate description

I was reading Gerald Edgar's "Measure, Topology, and Fractal Geometry" when I came across this exercise Let coordinates $(u,v)$ be defined in the plane with origin at one corner of the ...
Rubén Sales Castellar's user avatar
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Calculate the product of the segments described below

Points $A0, A1, A2.....A_{2n}$ divide a circle of radius whose size is $R$ into an odd number of congruent parts, $B$ is a point diametrically opposite to point $A0$. Calculate: $BA1. BA2. BA3. BA4. .....
peta arantes's user avatar
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Find the size of the radius of the nth circle.

In the figure shown, calculate the size of the radius of the nth circle. (Answer:$\frac{R}{n^2+2}$) I calculated the radius of the first two circles. How to extrapolate this resolution to other rays? ...
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Unit Disks problem

Good day, I'm faced with a problem that I don't understand how to approach. I would be grateful if you could tell me at least the direction in which to think. Three circles of unit radius are given ...
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Calculate the angle $x$ in the circle below

If $ \overparen{\mathit{MQN}}=240^\circ$, calculate $x^\circ$, with $M$ and $N$ being points of tangency. (Answer: $30^\circ$) I think: Let $O$ be the center of circle: $ \overparen{\mathit{MQN}}=240^...
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Find the radius $r$ on the circle tangent to two other circles

In the figure shown calculate $r$, if $R= 17m$ (Answer:$4$) I was able to calculate the radius of the circle on the right. How would you calculate the radius $r$? $ON = x\\ s = \frac{34}{4} = 8,5\\ \...
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Parallel line equation

I want to incorporate 2 diagonal lines in a logo design. The lines have to be parallel to each other and have to be exactly 0.5 inches apart when measured perpendicular. The upper point of Line 1 has ...
Geo's user avatar
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Find the segment "MD" in the internally tangent circles below [duplicate]

In the figure calculate $MD$. $BM = 6m$, $DE= 1m$. $BD \cap PQ = M$ I try: Let $N$ be the upper intersection of the segment $BE$ with the smaller circle $AD.DC = DE.BD = DE.(6+MD)$ $\therefore AD.DC =...
peta arantes's user avatar
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Find the segment $NQ$, in terms of $a$, $b$ and $c$, in the circle below

In the figure, if $CN = a$, $NH = b$ and $HQ = c$. Calculate $NQ$ if $F$, $N$, $Q$ and $P$ are points of tangency. (Answer: $c.\sqrt{\frac{a}{a+b}}$) I try: $J = CA \cap \odot(O_1,O_1F) \neq F$ note ...
peta arantes's user avatar
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Calculate the segment $MQ $in the circle below

In the figure if $RN = 5m$, $MB = 4m$ and $RN \parallel B$. Calculate MQ $\angle RMN = 90^o$ (Answer($\frac{4\sqrt{15}}{5}$) I try: $x+r = 4 \implies x= 4-r(1)$\ $AM=AB-BM =2r-4(2)$ $\theta = \frac{\...
peta arantes's user avatar
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Find the measurement of the $\angle PSQ$ in the semicircle below

If $\overset{\LARGE{\frown}}{PNQ} = 240^o $ find $\angle PSQ$ (Answer:$30^o $) I was unable to find much information for resolution; $\overset{\LARGE{\frown}}{PNQ} = 240^o \implies \overset{\LARGE{\...
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Find the lenght of segment $AE$ in the figure below

In the figure, $BC = 3$m and $CD = 1$m. Calculate $AE$. (Answer: $\sqrt6$) I try: $\triangle ABF: BF^2 = AB^2+AF^2 = AB^2+(2r+AE)^2$ $\triangle EFC: (2r)^2=EC^2+CF^2$ $AC.AH = AE.(AE+2r)$ $\triangle ...
peta arantes's user avatar
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Find the ray($x)$ of the quadrant below

In the figure if $AO = OB = x$, $AQ = 3$ and $BP = 4$. Calculate "$x$", where $MN^2 = AM^2+BN^2$ (Answer:5) I try: $BP^2 = BN.BO = BP.x = 16$ $AQ^2 = AM.x = 9$ $ \therefore \frac{16}{BP} =\...
peta arantes's user avatar
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Find the value of the tangent segment PC in the circle below

In the figure if PH = 15m, HT = 8m and the radius of the semicircle is 13 m. Calculate PC with P and T tangency points. (Answer:$\frac{13\sqrt3}{3}$) I try: $\triangle PCO: OC^2 = CP^2+PO^2 = \...
peta arantes's user avatar
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Calculate the segment $AC$ in the secant circles below

In the figure, $AR =a$, and $CS = b$. Calculate $AC$, with $M$, $N$, $R$ and S points of tangency. (Answer:$\sqrt{a^2+b^2)}$ Itry: Let $R$ be the radius of the largest circle and $r$ of the smallest $...
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Find the segment BT in the triangle inscribed below

In the figure, $AB.BC = 60$ and $BT.TP = 40$. Calculate BT with B and T tangency points. (Answer:$2\sqrt5$) I try: $AT.TC = BT.TP \implies AT.TC = 40$ $AM.AB = AT.AC$ $AT^2 = AM.AB \implies AT^2 = AT....
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construct series of planar polygonal spirals that approximate planar curve

Let $\gamma:[a,b]\rightarrow \mathbb{R}^2$ a constant speed spiral, that is a) $\gamma$ is locally convex w.r.t. 0, that is $\gamma$ has supporting lines that locally lie above $\gamma$ and $\gamma$ ...
Mathemann's user avatar
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Find the segment BT in the right triangle below

In the right triangle $ABC$, $AE = 9m$ and $EB = 3m$. Calculate $BT$.(Answer: 6m) I try: $BT^2 = BM.BC = BM.(BM+2R)$ $BT^2 = BN.BF$ $\triangle ABC: 12^2+ BC^2 = AC^2$ T.Menelaus: $\triangle ABC-BF: \...
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In the figure if calculate the $DM$ segment

In the figure if $AB = 3$m and $BC = 2m$ Calculate $DM$.(M is point of tangency) (Answer: $5\sqrt6$) I try: $DM^2 = DC(DC+5) = DC^2+5DC$(I) $DC.(DC+5) = DF.(DF+2R) \implies DC^2+5DC = DF^2+2DF.R(II)$ ...
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Find the size of the segment between $2$ points belonging to two tangents described below

A quadrilateral $ABCD$ inscribed in a circumference, the extensions of $AB$ and $DC$ intersect at point $"P"$ and the extensions of $BC$ and $AD$ intersect at point $"Q"$. Calculate the length of $PQ$,...
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Prove a pair of edges cannot intersect more than once if the drawing has $\mbox{cr}(G)$ crossings

I am trying to prove that if a drawing of a graph $G$ has $\mbox{cr}(G)$ edge crossings, then any pair of edges don't intersect more than once in the drawing. Here, $\mbox{cr}(G)$ is the crossing ...
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Find the sum of the sizes of the segments formed by the Feuerbach point and the Nagel point

In an acute-angled triangle, if the sizes of the segments that join the median point with the Spieker point, the Spieker point with the circumcenter and the median point with the circumcenter are $5$,$...
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Calculate the radius of the circle that is tangent to the two others and to a line tangent to these two circles

Two circles with radii $R$ and $r$ intersect at an angle of $120°$. A common external tangent $AB$ is drawn ($A$ and $B$ are points of tangency). Calculate the size of the radius of the circle that is ...
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Rigorous proof of ASA Congruence rule

Using Hilbert's axioms of geometry how can we prove with rigor that the angle-side-angle congruence rule is true? Most of the proofs I have seen of this seem to rely on the diagram and follow this ...
MushroomTea's user avatar
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Find the sum $PC^2+PD^2$ in the trapezoid inscribed below

If there is a semicircle of diameter $AB$ in which an isosceles trapezoid $ABCD$, ($AB \parallel CD$) is inscribed. On $AB$, we take a point "$P$" such that $PA^2 + PB^2 = 5^2$. Calculate: $...
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Find the distance from the apex of a trapezium to the midpoint of the opposite side

If you have a trapezium $ABCD$ ($BC\parallel AD$) whose average base measures $2$m. Calculate $DM$ with $"M"$ being the midpoint of $AB$, in addition $CD^2-2.MC^2 = 2m^2$. (Answer:$3$) I made a ...
peta arantes's user avatar
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Find is the value of the radius "x" in the circle below as a function of R and r

What is the value of the radius $"x"$ in the circle below? (Answer: $\frac{r}{R}(R-r)$) I try: $\triangle OMN:(R-2r)^2+MN^2=R^2$ $R^2-4Rr+4r^2+MN^2=R^2$ $MN^2=4Rr-4r^2$ $MN^2 = 4r(R-r)$ I can't see ...
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Geometric construction involving a line crossing a square with a given length

ABCD is a unit square. Construct a point E on DC (extended) such that AE intersects BC at F with EF = 1. After trying for long, the only solution I could find was to solve for the length of $CE$ ...
Soham Saha's user avatar
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FInd the distance between the centers of the circles

In the figure shown, $O$ and $O_1$ are the centers of the circles. If $r=6$m and $R=8$m, Calculate the size of $OO_1$(Answer:$10$m) I think that the upper point of the intersection of the circles ...
peta arantes's user avatar
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Find the segment $x$ in the semicircle below

In the figure shown, if $PM = MQ = 4$, calculate $x$. $O$ and $O'$ are centers. (Answer:$x=4$) I try: $\triangle OQA: (OP+8)^2 = OA^2+AQ^2 =OA^2 +(AC+CQ)^2$ $\triangle OCB: AC^2 = AO.AB$ $\triangle ...
peta arantes's user avatar
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Find the side of the square ABCD in the figure below

In the figure, ABCD is a square. If $\dfrac{1}{BM^2}+\dfrac{1}{BN^2} = \dfrac{1}{25}$, calculate AB (Answer: $AB=5$) I try: $\triangle ABN \sim \triangle DMN: \frac{MN}{BN} = \frac{MD}{L}=\frac{ND}{L+...
peta arantes's user avatar
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Find the relationship between a, b and c in the figure bellow.

In the figure, if $\overset{\LARGE{\frown}}{MN} = 90^o$, calculate the relationship between a, b and c. (Answer: $b^2=a^2+c^2$) I try: $HP=r_1: OP=x_1: DQ=r_2 :OQ=x_2: OM=r\\ \triangle OPH :(R−r_1)^...
peta arantes's user avatar
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2 votes
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Calculate the $PQ$ segment in the semicircle below

In the figure, if $MN = 40$ and $NQ = 9$, Calculate $PQ$. $AB$ and $MB$ are diameters. (Answer: $21$) Note: The question does not mention whether the diameters are equal. I don't know if it would be ...
peta arantes's user avatar
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4 votes
4 answers
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Find the radius $r$ in the circle below

In the figure, $OP = 1$ and $PF = 3$. Calculate $r$. $DC$ and $DE$ are tangents. (Answer:$r=2$) Could this issue be resolved only with this data? I try: $Point Theorem: FA.FB = FC.FE \implies (3-(r-1)...
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1 answer
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Find the size of the radius of the circle.

In the figure. if: $AB = 12m$; $AC = 4m$ and $HE = 8m$. Calculate the size of the radius of the circle.(Answer:$R=8$) I try: $\triangle ABD: BE^2 =AE.ED =(R+OE).(R-OE) = R^2-OE^2$\ $\triangle ABD: BE....
peta arantes's user avatar
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2 votes
1 answer
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Converting 3d coplanar points to space of 2d plane then back into 3d

I have the equation of an arbitrary plane of the form $Ax+By+Cz+d= 0$. I also have a set of points lying on this plane, $\{p_1, p_2, ... , p_i\}$ where each $p_i$ is an $(x,y,z)$ coordinate. I would ...
wkacct acctwk's user avatar
1 vote
1 answer
150 views

Find the area of ​triangle ABC depending on the conditions given below

In triangle $ABC$, angle $C$ measures $120°$ and side $AC$ is greater than side $BC$. Knowing that the area of ​​the equilateral triangle with side $AB$ is $31$, the area of ​​the equilateral triangle ...
peta arantes's user avatar
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2 votes
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Determining the Plane for a Given Point and Line Using the Equation of a Plane

I was working on an exercise following an example that required exactly what I had to do as well, but I encountered some difficulties. Below, I have indicated 2 titles, the first one is the example I ...
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