 # High school geometry questions and answers

Recent questions in High school geometry Paul Duran 2022-05-09 Answered

### Show that $f\left({x}_{1}^{\ast },...{x}_{n}^{\ast }\right)=max\left\{f\left({x}_{1},...,{x}_{n}\right):\left({x}_{1},...,{x}_{n}\right)\in \mathrm{\Omega }\right\}$ if and only if $-f\left({x}_{1}^{\ast },...{x}_{n}^{\ast }\right)=min\left\{-f\left({x}_{1},...,{x}_{n}\right):\left({x}_{1},...,{x}_{n}\right)\in \mathrm{\Omega }\right\}$I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of $f$. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts? Micah Haynes 2022-05-09 Answered

### Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.These $n$ points divide the circle into $n$ arcs.Let ${Z}_{i}$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.What is $E\left[X\right]$ and $Var\left[X\right]$?Any hints will be appreciated. Thanks..(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal) Alexis Meyer 2022-05-09 Answered

### Triangle law of vector addition vs Pythagorean theoremSuppose there is a vector a of magnitude 5 units to the east, another vector b of magnitude 6 units to the north. To find magnitude of vector a + vector b,By the triangle law of vector addition, it is 5 + 6 units = 11 units.By Pythagorean theorem, it is $\sqrt{\left(}{5}^{2}+{6}^{2}\right)=\sqrt{\left(}61\right)$Which answer is right? If so, why is the other wrong?Thank you! adocidasiaqxm 2022-05-08 Answered

### The length of an arc of a circle is 12 cm. The corresponding sector area is 108 cm${}^{2}$. Find the radius of the circle.I have not attempted this question and do not understand how to solve this. Carina Valenzuela 2022-05-08 Answered

### I want to solve the following optimization problemwhere the design variable $X$ is symmetric positive semidefinite, $W,Y$ are fixed symmetric positive semidefinite matrices, and $a$ is a given vector. The inequality $X⪰W$ means that $X-W$ is symmetric positive semidefinite.I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically? lurtzslikgtgjd 2022-05-08 Answered

### Find the length of base of a triangle without using Pythagorean TheoremI'm curious whether it is possible to find the length of base of the triangle without using Pythagorean TheoremNo Pythagorean Theorem mean:=> No trigonometric because trigonometric is built on top of Pythagorean Theorem. etc $\mathrm{sin}\theta =\frac{a}{r}$=> No Integration on line or curve because the integration is built on top of Pythagorean Theorem. etc: $s\left(x\right)=\int \sqrt{{f}^{\prime }\left(x{\right)}^{2}+1}$ Azzalictpdv 2022-05-08 Answered

### Let be $t\in \mathbb{R}$, $n=1,2,\cdots$, $p\in \left[0,1\right]$ and $a\in \left(p,1\right]$.Show that$\underset{t}{sup}\left(ta-\mathrm{log}\left(p{e}^{t}+\left(1-p\right)\right)=a\mathrm{log}\left(\frac{a}{p}\right)+\left(1-a\right)\mathrm{log}\left(\frac{1-a}{1-p}\right)$So, I've try to derivate, but I did'nt get sucess, since my result is different. I've get $\mathrm{log}\left(\frac{a\left(1-p\right)}{p\left(1-a\right)}\right)$. Any ideas? measgachyx5q9 2022-05-08 Answered

### Suppose we have some functions ${f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{n}\left(x\right)$ with $x\in {\mathbb{Z}}^{n}$.We can denote the subset ${X}_{1}$ of ${\mathbb{Z}}^{n}$ that maximizes ${f}_{1}\left(x\right)$ as:${X}_{1}=\underset{x\in {\mathbb{Z}}^{n}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}\left(x\right)$Now, suppose there is a kind of "priority" in which I also want to maximize ${f}_{2}$, as long as I keep maximizing ${f}_{1}$. This could be represented as:${X}_{2}=\underset{x\in {X}_{1}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{2}\left(x\right)$The same for ${f}_{3}$:${X}_{3}=\underset{x\in {X}_{2}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{3}\left(x\right)$So on and so forth...Is there some, more concise, notation to represent this "maximization priority"? poklanima5lqp3 2022-05-08 Answered

### The problem is like$ma{x}_{\mathbf{x}}\phantom{\rule{mediummathspace}{0ex}}u\left({x}_{1},{x}_{2},...,{x}_{L}\right)=-\sum _{i=1}^{L}\mid {x}_{i}-{a}_{i}\mid$,$s.t.\sum _{i}^{L}{x}_{i}\le C$for each $i$, ${a}_{i}>0$ is a scalar;$C$ is a constant that is strictly greater than 0;$\mathbf{x}=\left({x}_{1},{x}_{2},...,{x}_{L}\right)\in {\mathbf{R}}_{+}^{L}$. Characterize the optimal $\mathbf{x}$ as a function of $C$ or ${a}_{i}$.Hint: to solve the problem we should discuss the cases when $C\le \sum _{i}{a}_{i}$ and $C\ge \sum _{i}{a}_{i}$.Thank you! encamineu2cki 2022-05-08 Answered

### Let $X\in \left\{0,1{\right\}}^{d}$ be a Boolean vector and $Y,Z\in \left\{0,1\right\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y,Z$ and we'd like to find a joint distribution ${\mathcal{D}}^{\prime }$ over $X,Y,Z$ such that:1. The marginal of ${\mathcal{D}}^{\prime }$ on $Y$,$Z$ equals $\mathcal{D}$.2. $X$ are independent of $Z$ under ${\mathcal{D}}^{\prime }$, i.e., $I\left(X;Z\right)=0$.3. $I\left(X;Y\right)$ is maximized,where $I\left(\cdot ;\cdot \right)$ denotes the mutual information. For now I don't even know what is a nontrivial upper bound of $I\left(X;Y\right)$ given that $I\left(X;Z\right)=0$? Furthermore, is it possible we can know the optimal distribution ${\mathcal{D}}^{\prime }$ that achieves the upper bound?My conjecture is that the upper bound of $I\left(X;Y\right)$ should have something to do with the correlation (coupling?) between $Y$ and $Z$, so ideally it should contain something related to that. Jay Barrett 2022-05-08 Answered

### Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as $y=b-ax$, and $y=c-d{x}^{2}$, but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance. Matthew Hubbard 2022-05-08 Answered

### Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using arc lenghts. Even though they seemed to be on the right track, there is a problem:They are using sinx and cosx functions, which are actually defined using the perimeter of a circle, aren't they? So, in my opinion, that was not a real proof since they were using what they want to show.Or am I missing something? Was that a real proof?Edit: I don't know but most probably they defined sinx and cosx in the usual way, because it was a calculus class. hyprkathknmk 2022-05-08 Answered

### How did Pythagoras argue for the Pythagorean theorem?There are lots of proofs for the Pythagorean theorem. But, how did Pyhtagoras argue for the theorem himself? Logan Lamb 2022-05-08 Answered

### Which are the most famous problems having an objective of maximizing a nonlinear convex function (or minimizing a concave function)? As far as I know such an objective with respect to linear constraints is np-hard. arbixerwoxottdrp1l 2022-05-08 Answered

### Let $X=\left({X}_{1},...,{X}_{n}\right)$ be a vector of $n$ random variables. Consider the following maximization problem:$\underset{a,b}{max}\phantom{\rule{thickmathspace}{0ex}}\mathrm{C}\mathrm{o}\mathrm{v}\left(a\cdot X,b\cdot X\right)$ under the constraint that $‖a{‖}_{2}=‖b{‖}_{2}=1$.($a\cdot X$ is the dot product between $a$ and $X$). Would it be true that there is a solution to this maximization problem such that $a=b$?Thanks. syaoronsangelhwc17 2022-05-08 Answered

### An airline will fill 100 seats of its aircraft at a fare of 200 dollars. For every 5 dollar increase in the fare, the plane loses two passengers. For every decrease of \$5, the company gains two passengers. What price maximizes revenue?I'm a bit lost as to what my equations should be. The total revenue, if all seats are filled can be 20000. So for every 5 dollar increase the total revenue becomes: 20000−400x (400x = 2 people time the price of each ticket times the number of 5 dollar increases). I don't know how to model the other situation of a 5 dollar decrease. lasquiyas5loaa 2022-05-08 Answered

### How to state Pythagorean theorem in a neutral synthetic geometry?In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence one has first to state the pythagorean theorem in neutral geometry (I name 'neutral geometry' a geometry in which parallel lines do exist but with the parallel postulate removed).If one start with an axiom system like Birkhoff's postulates which assume reals numbers and ruler and protractor from the beginning then there is no problem stating the Pythagorean theorem.My question is how one can one state the Pythagorean theorem in a neutral synthetic geometry based on axioms such as Hilbert's axioms group I II III or Tarski's axioms A1−A9 ?It is possible to define segment length in neutral Tarski's or Hilbert's geometries as an equivalence class using the congruence (≡) relation. It is also possible to define the congruence of triangles.However, the geometric definition of multiplication as given by Hilbert assume the parallel postulate. The existence of a square is equivalent to the parallel postulate. enclinesbnnbk 2022-05-07 Answered

### Let $x\in {\mathbb{R}}^{n}$ and let $P$ be a $n×n$ positive definite symmetrix matrix. It is known that the maximum of$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& {x}^{T}x\le 1\end{array}$is ${\lambda }_{\text{max}}\left(P\right)$, the largest eigenvalue of $P$. Now consider the following problem$\begin{array}{ll}\text{maximize}& {x}^{T}P\phantom{\rule{thinmathspace}{0ex}}x\\ \text{subject to}& \left(x-a{\right)}^{T}\left(x-a\right)\le 1\end{array}$where $a\in {\mathbb{R}}^{n}$ is known. What is the analytical solution? Flakqfqbq 2022-05-07 Answered

### "A common theme in linear compression and feature extraction is to map a high dimensional vector $x$ to a lower dimensional vector $y=Wx$ such that the information in the vector $x$ is maximally preserved in $y$. Opten PCA is applied for this purpose. However, the optimal setting for $W$ is in generall not given by the widely used PCA. Actually, PCA is sub-optimal special case of mutual information maximisation."Can anyone elaborate why PCA is a sub-optimal special case of mutual information maximisation ? Justine Webster 2022-05-07 Answered

### Some have proposed that for a natural centrality measure, the most central a node can get is the center node in the star network. I've heard this called "star maximization." That is, for a measure $M\left(\cdot \right)$, and a star network ${g}^{\star }$ with center ${c}^{\star }$,$\left\{\left({c}^{\star },{g}^{\star }\right)\right\}\in {\mathrm{arg}max}_{\left(i,g\right)\in N×\mathcal{G}\left(N\right)}M\left(i,g\right)$where $N$ is the set of nodes and $\mathcal{G}$ considers all unweighted network structures.I'd like to learn about some centrality measures that don't satisfy this property, but "star maximization" isn't a heavily used term, so I am having trouble in searching for many such measures. What are some such measures of centrality?

It will include more complex tasks to solve like congruence equation solving, which also becomes easier when you have a good example like what we provide.