High school geometry questions and answers

Recent questions in High school geometry

2022-04-07
2022-04-06

Consider the example of maximizing ${x}^{2}yz$ under the constraint that ${x}^{2}+{y}^{2}+{z}^{2}=5$.One way to do this is to use lagrange multipliers, solving the system of equations$2xyz=2x\lambda$${x}^{2}z=2y\lambda$${x}^{2}y=2z\lambda$${x}^{2}+{y}^{2}+{z}^{2}=5$However, couldn't you just substitute ${x}^{2}=5-{y}^{2}-{z}^{2}$ into the expression you want to maximize to get: $yz\left(-{y}^{2}-{z}^{2}+5\right)$and then just maximize that by setting the $y$ and $z$ partial derivatives equal to zero?Then you just have to solve the arguably simpler system of equations:$3{y}^{2}z+{z}^{3}=5z$${y}^{3}+3y{z}^{2}=5y$where the $z$s and $y$s cancel out nicely on both sides.Why is maximize by lagrange multipliers necessary when you can always substitute and maximize the resulting function?When should you choose one over the other?

Hailee Stout 2022-04-06 Answered

It is not possible for a part of any of three conic sections to be an arc of a circle. It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?

Karissa Sosa 2022-04-06 Answered

Suppose I'm given two points: $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those two points (the arc with shorter length)?

Eve Dunn 2022-04-06 Answered

I need help with the following optimization problem$max\phantom{\rule{thickmathspace}{0ex}}\alpha \mathrm{ln}\left(x\left(1-{y}^{2}\right)\right)+\left(1-\alpha \right)\mathrm{ln}\left(z\right)$where the maximization is with respect to $x,y,z$, subject to$\begin{array}{rl}\alpha x+\left(1-\alpha \right)z& ={C}_{1}\\ \alpha y\sqrt{x\left(x+\gamma \right)}-\alpha x& ={C}_{2}\end{array}$where $0\le \alpha \le 1$, $\gamma >0$, and $x,z\ge 0$, and $|y|\le 1$.Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

briannalofton27 2022-04-01
2022-03-29
Jewel Dacumos 2022-03-29 Answered

Determine whether each statement is true or false. If it is false, provide a counterexample. 1.If a quadrilateral is equilateral, then it is equiangular. 2. If a quadrilateral is equiangular, then it is equilateral. 3. Every isosceles triangle has at least one line of symmetry. 4. Every equilateral triangle has exactly three lines of symmetry. 5. If two angles are not congruent, then the two angles are not right.

Deegan Chase 2022-03-28 Answered