# Tagged Questions

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### How can I find the volume of prism: $V = \frac{(a + b + c)Q}{3}$

In the book Handbook of Mathematics (I. N. Bronshtein, pg 194), we have without proof. If the bases of a triangular prism are not parallel (see figure) to each other we can calculate its volume by ...
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### Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
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### To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
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### can you help me solve my menu board dilemma?

If I have a menu board that measures 35 3/8 $\times$ 71 5/8 and I need to cut in 3 equal pieces, what measurements should each piece be?
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### Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
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### Rotation of an equation

Problem: Suppose a line $L$ is given by the equation $\frac{x}{a} + \frac{y}{b}=1$, where $a$ and $b$ are non-zero real numbers. Let $\Re_{\frac{\pi}{2}}$ be the counterclockwise rotation of the ...
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### Proof about isometries, symmetry and reversing orientation.

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry of the reals. Prove that $f$ is a symmetry around a point if and only if $f$ reverses orientation of $\mathbb{R}$. The orientation of ...
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### Proof about symmetry in isometries.

Suppose $f: \Bbb R \rightarrow \Bbb R$ is an isometry of the reals. Prove that $f$ is a symmetry about a point if and only if $f$ has a unique fixed point. Part 1: The assumption is $f$ is a ...
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Suppose $f\colon\mathbb R\to\mathbb R$ is an isometry of the reals. Prove $f$ is a non-trivial translation iff $f$ has no fixed points. Assumption: $f$ is a non-trivial translation (trivial ...
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### Plane isometries $g\ , f$ , properties of fixed points and types

Given two plane isometries $g\ , f$ and $f^{'} = g\circ f \circ g^{-1}$ prove that: If $P$ is a fixed point of $f$ then $g\left(P\right)$ is a fixed point of $f^{'}$ and if $Q$ is a fixed point of ...
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### What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?

Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)? Okay i have tried couple of things but seems its not working . Please suggest me proper ...
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### Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
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### 3D Geometry Question

In $3$-dimensional Geometry, if angle made of line segment $OP$ with $X,Y,Z$-axis are in $1:2:3$, then what is the angle made by line segment with $Y$-axis? My Solution: Let $\alpha,\beta$ and ...
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### Problem on hyperbolic hyperboloid generated by a rotation

This is the problem: In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
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### A question on Trigonometry (bisector)

If two bisector of a triangular is equal, then it is Isosceles triangular.
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### prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
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### coordinate geometry: finding the ratio in which a line segment is divided by a line

The question is: Determine the ratio in which the line 3x + 4y - 9 = 0divides the line segment joining the points ...
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### 6 point lying on a common circle

$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
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### Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$

Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$. I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
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### Questions related to vectors and linear algebra

1) Given vectors $u = (1, 1, -2)$ and $v = (0, 1, 1)$, find the value of $t$ such that magnitude of $u + t(v)$ has the smallest value I have no idea how to begin. 2) Given vectors $u = (1, 1, -2)$, ...
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### Euclidean Conjugation group

Having a tad bit trying to prove this question, Show that the set of reflections is a complete conjugacy class in the euclidean group E. Also, do the same for the set of half-turns and inversions. ...
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### there are two docks

There are two docks, dock A and Dock B, on a large lake. The distance between the two docks is 72.5 km. Dock B is directly east of dock A. One day, a steam boat leaves from dock A at noon, and heads ...
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### Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
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### Euclidean geometry exercise

I would like some help to solve this: Consider a triangle $\triangle ABC$ with $\angle A$ a right angle and $BC=20$. Divide $BC$ into four congruent segments, that is, take the points ...
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### Pythagorean theorem

We can make a square into four equal squares. Fine, if we want to make into five.. Then there is a problem. Please discuss, How to make five squares from a single square by using a Pythagorean ...
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### Represent lengths rectangle using given terms

In a rectangle, $GHIJ$, where $E$ is on $GH$ and $F$ is on $JI$ in such a way that $GEIF$ form a rhombus. Determine the following: $1)$ $x=FI$ in terms of $a=GH$ and $b=HI$ and $2)$calculate $y=EF$ in ...
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### Prove that intersection of diagonals in trapezium divide parallel segment to equal segments

I've got exercise to do as en exercise to my school leaving exam and I have no idea how to prove it: Diagonals of trapezium intersect in point $S$. Through point $S$ the segment was given that ...
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### Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...
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### Relationship Between Three Incircles

Suppose we have a right triangle $ABC$ with hypotenuse $AB$. Further, suppose the altitude from $C$ hits hypotenuse $AB$ at point $D$. If the inradius of $ACD$ is $r_1$ and the inradius of $BCD$ is ...
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### Systems of equations finding right triangles

I need help setting up the equation for the question, "Find all right triangles for which the perimeter is $24$ units and the area is $24$ square units." I know that the area is $A = \frac12 b h$ ...
### Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$
Let there be a circle $(O,R)$ and $AB,CD$ two perpendicular chords of that circle that intersect on point $E$. Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$