# Tagged Questions

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### Show an immersion is locally one to one using the inverse function theorem

Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint?
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### Why do we use $cm^2$?

I can't seem to wrap my head around why we should use $cm^2$ for area. According to my textbook we use it for converting units of area but I don't understand how $1cm$ is any different from $1cm^2$. ...
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### Find out the $\angle PRQ$

please, help me to solve this.How can I proceed.I just need help. $PQR$ is a triangle. $M$ is a point on $QR$.here,$QM=1/3RM$ , $\angle RPM=30^ \circ$ and $\angle QPM=20^ \circ$ now,$\angle PRQ=??$ ...
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### Calculating Angles at Vertices

This is quite a tricky question and I can't answer it.
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### Sketching coordinates of C when it meets the x and y axis

The question: Curve $C$ has the equation $y=(x-k)^2 (x-k+2)$ Where $k$ is a constant and $k > 2$. Sketch $C$, showing the coordinates of the point where $C$ meets the $x$ and $y$ axes. I am ...
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### Geometry: Using Pythagorean Theorem to find lengths of triangle

I am helping someone do their geometry homework, and wanted to make sure I am even doing it correctly. Below is the problem: Here is how I solved it: We know B is the center, so AB=BE. ...
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### Geometry: Find the angle x

I am trying to help my little sister do her geometry and seem to have forgotten my basic math skills. Here are a couple she sent me: Any help would be great! Once I get back into the grove I ...
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### co-ordinate geometry equations

The center of a circle s is the second quadrant. s touches both the x-axis and the y-axis at the points P and Q respectively s has a radius length of 5 root 2. i. find the equation of s ii. Find the ...
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### Finding the measure of an arc on a circle

If someone could work me through how to solve this, that would be great because I am stumped on this one. I know it looks like there is a lot of useless information in the picture, but there are ...
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### The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
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### Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
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### Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
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### What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
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### Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
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### Right triangles, minimum difference of two measured angles

The question reads: If $\angle ABC$ is a right angle and the measure of $\mu(\angle ACB)=r^{\circ}$, what is the minimum difference between $\mu(\angle ACD)$ and $\mu(\angle BAC)$? Which of the three ...
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### Prove that the angle is 45

In ΔABC, ∠B is a right angle. D and E are points on segment AC such that AD:DE:EC = 1:2:√3. Then, prove that m∠DBE = 45.
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### Given A and B two fixed points on the circle find the locus of the orthocenter of triangle ABC where C is a mobile point on the circle.

First we have to find the locus and then we have to prove double inclusion. Please help me !
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### Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
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### Calculate angles of a projection of a tetrahedron

ABCD is a regular tetrahedron. It is projected on a plane in such a way that the projection forms an isosceles triangle ABC (AB = BC ≠ AC) with D lying in the middle of AC (right image): Problem: ...
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### Geometry problem without using trigonometry or similar triangles.

I was helping my sophomore friend with her geometry homework and came to problem 13a: My first instinct was to use similar triangles and say $\triangle BKC \sim \triangle CMD$, and from there we ...
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### Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
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### In sphere $r \propto \frac{1}{A}$! How is this possible? What's the wrong here?

Surface area $A$ and volume $V$ of a sphere of radius $r$ are \begin{eqnarray} A=4\pi r^2,\\ V=\frac{4}{3} \pi r^3. \end{eqnarray} But then \begin{align} \frac{V}{A} & = \frac{r}{3}\\ ...
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### Question about relationship between line and plane in solid geometry?

Given: Cube ABCD.EFGH, with P and Q as midpoints of CG and BF respectively. What is the relationship between: a. EP and plane ABCD b. AC and plane EQPH My attempt: a. Since EP at plane ACGE ...
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### How to draw a intersection of two cones graphically

What is the best software we can use to draw the intersections of two cones such as ellipses and hyperbolas mathematically and clearly. Please send some drawaings and software you have used?
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### Uniqueness of Point

I want to proove: For points $P \neq Q \in \mathbb{R}$ and $a,b >0$ there is exactly one point $X \in [PQ]$ and exactly one point $X' \notin [PQ]$ with ratio $a:b$. For $a=b$ there is only one ...
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### Do I Have To Explicitly Define Points/Lines/Planes? [closed]

The problem I was given is as follows: Assignment: Draw and label the following figures. Two planes that do not intersect. Lines LM and NP on the same plane (coplanar) but do not ...
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### Distance of convex combination of pairs of points in $\mathbb{R}^n$

Given 4 points $w,x,y,z \in \mathbb{R}^n$ define for $t\in [0,1]$ $f(t)=d(wt + (1-t)x, yt + (1-t)z)$. Is this function convex? I have found a proof by differentiating twice and calculating a lot but ...
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### How to prove that certain points relating to a trapezoid are collinear?

Can you help me to prove that in any trapezoid, which is not a parallelogram, the following points are collinear? The midpoints of its bases. The point of intersection of diagonals. The point of ...
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### Finding ratio of external division in a triangle.

Given a $\triangle ABC$ and $P$ dividing $AB$ internally in the ratio $2:3$ $Q$ dividing $AC$ internally in the ratio $1:2$ , with $PQ$ produced and $BC$ produced intersecting in $R$ , to find the ...
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### Given the length of two altitudes and one side , find the area of triangle.

Segments $BE$ and $CF$ are the altitudes in $\triangle ABC$. $E$ is on line $AC$ and $F$ is on line $AB$. $BC = 65$, $BE = 60$ and $CF = 56$. Find $A(\triangle ABC)/100$. By the Pythagorean ...
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### In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA= 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA = 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$. Here is a sketch that I drew: From the given ...
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### Estimating Eigenvalues and Eigenvectors from an 'eignpicture'

I was given this question at my school but it really does not make sense to me: The unit vectors $x$ in $\mathbb{R}^2$ and their images $Ax$ under the action of a $2x2$ matrix A are drawn ...
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### A circle is inscribed in sector of another bigger circle.Given A(circle) find the A(triangle formed by the center and the endpoints of the sector).

Consider sector of circle $MAB$. $∠AMB = 120◦$. A circle $S$ touches side $AM$, side $MB$ and arc $AB$ as shown in the ﬁgure. Area of circle $S$ is $75π/(7 + 4√3)$ . Find $4√3$ times the area of ...
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### Right -angled triangle perimeter question

A right-angled triangle has an area of 5. The altitude perpendicular to the hypotenuse has a length of 2. Calculate the perimeter of the triangle. I could not get the area of the triangle to be 5 so ...
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### Minimum distance to points in plane

Someone told me that the the following problem is elementary. Given three points $a=(-5,0)$, $b=(0,5)$ and $c=(5,0)$ in $\mathbb R^2$ with Euclidean norm: \mbox{minimize}\;\; \; f(x)=\|x-a\| + ...
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### Determine the angle knowing following…

I have such a problem: In triangle ABC two medians AD and BE intersect in point O. Determine the measure of the angle BAC knowing that triangle AOE is equilateral. I've drawn such a triangle and I ...
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### A question related to triangles , areas , ratio of areas of triangles.

I know the title is confusing but that is because of 150-character limit, if anyone of you can improve it , please do. Consider $\triangle ABC.$ Choose a point $D$ on segment $BC$ such that ...
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### Geometry problem with 2 circles and a triangle

I tried to solve this problem: But I did not know how to do it so I looked at the answers and I saw E looked convincing because it is the only one that has square powers and D (from the diagram) is ...
We have given three points in the Euclidean space such as $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and $C=(x_{2},y_{2},z_{2})$ and point $D$ moves inside the triangle $\triangle ABC$ such ...