High School Geometry Examples and Practice Questions

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High school geometry

Recent questions in High school geometry

Shamar Reese 2022-05-28

Suppose we have a primal problem $P$ which is stated as a maximization problem $max c^{T} x$ .

The dual problem is defined only for a primal minimization problem.

Then what is the dual problem of $P$ ?

Is it implicit, that the dual problem of $P$ is the dual problem of $P$ stated as the minimization problem $min - c^{T} x$ ?

Surely these two primal problems are equivalent in the sense that their optimal solution $\bar{x}$ are equal (if it exist). However, the objective values are the same only if we ignore the sign !

Quintacj 2022-05-28

Minimize the expression:
$F (a, b) = a b + a b^{'} + a^{'} b$ .
(A) $a^{'} + b^{'}$
(B) $a^{'} + b$
(C) $a + b$
(D) $a + b^{'}$
I have no idea how to solve this problem. Also how to maximize the expression and what is the value of maximization.

dglennuo 2022-05-28

Solve for x in
$\frac{\sin (30) \cdot \sin (60) \cdot \sin (x + 10)}{\sin (80) \cdot \sin (40) \cdot \sin (x)} = 1$

Xiomara Poole 2022-05-28

Having the equation
$a^{T} x = b$
where $a$ , $a, x \in R^{n}$ , $b$ is a number. Can I solve $x$ in terms of $a$ , $b$ ?

Ultimate goal is to find the maximum of
$\frac{1}{2} x^{T} P x$
given $a^{T} x = b$ , where $P$ is a symmetric positive definite matrix.

Jaycee Mathis 2022-05-28

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done...
$M a x \frac{1}{| | x | |}$
is converted into
$M i n \frac{1}{2} x^{T} x$
How was this step achieved..? Many thanks in advance !

Alaina Marshall 2022-05-28

How does the attached figure prove the Pythagorean theorem?

Jerry Villegas 2022-05-27

I have no idea how to solve the following question and will be grateful for any help. Suppose we are given $n$ real numbers $y_{1}, . . ., y_{n}$ . Is there any simple way to minimize function $f (y) = \sum_{k = 1}^{n} | y_{k} - y |$ ?

Hayley Sanders 2022-05-27

How to find intersection point of lines
$p . . . \frac{x + 1}{0} = \frac{y - 2}{- 1} = \frac{z}{1}$ and $q . . . \frac{x - 1}{- 1} = \frac{y + 6}{3} = \frac{z + 6}{4}$ ?

Alaina Marshall 2022-05-26

The primal assignment problem that I have is:
$\begin{aligned} max \sum_{i = i}^{N} \sum_{j = 1}^{N} c_{i j} x_{i j} \\ subject to \\ \sum_{i = 1}^{N} x_{i j} = 1, j = 1, \dots, N \\ \sum_{j = 1}^{N} x_{i j} = 1, i = 1, \dots, N \\ x_{i, j} \in {0, 1} \end{aligned}$
and I would like to obtain its dual, which should be
$min_{p_{j}} {\sum_{i = 1}^{N} p_{j} + \sum_{i = 1}^{N} max {a_{i j} - p_{j}}}$
Can you give me any hints on how to approach this derivation? I have tried with Lagrange multipliers but I have not been able to reach this expression.

istremage8o 2022-05-26

What is the equation of reflection of the curve
$y = 5 x^{2} - 7 x + 2$
about the pont (3, -3)?

Ryker Stein 2022-05-26

A cylindrical hole of radius xis bored through a shere of radius R in such a way that the axis of the hole passes throiugh the centerof the sphere. Find the value of x that maximizes the complete surface area of the remaining solid. Hint: The area of a segment of height h on a shere of radius R is $2 π R h$

istremage8o 2022-05-25

Given $Ω := [x_{a}, x_{b}] \times [y_{a}, y_{b}] \subset R^{2}$ , consider the problem
$max_{f \in C^{2} (Ω, R)} \iint_{Ω} [f_{y} (x, y) \int_{x_{a}}^{x} f (z, y) d z] d x d y$
subject to a constraint of the form
$\iint_{Ω} {[\int_{x_{a}}^{x} f_{y} (z, y) d z - \int_{x_{a}}^{x_{b}} z f_{y} (z, y) d z]}^{2} d x d y = c \in R$
where the subscript $y$ denotes the partial derivative w.r.t. the second variable, i.e., $y$ .

I thought the first step would have been to write down the Euler-Lagrange equations, i.e., given the Langragian $L$ ,
$\frac{\partial}{\partial y} (\frac{\partial L}{\partial f_{y}}) = \frac{\partial L}{\partial f}$
but the Lagrangian is pretty involved and I got stuck. Do you have any hint?

ASHLYN OWENS2022-05-24

the point (3,-2) is rotated 90 about the origin and then dilated by a scale factor of 4. what are the coordinates of the resulting image

shelohz0 2022-05-23

Analytic functions on spaces over non-Archimedean fields and troubles with totally disconnectedness. Why totally disconnectedness of the spaces causes big troubles if one tries to study the space using naively defined analytic functions on it locally expressible in power series?

Chaz Blair 2022-05-21

Given a point on a hyperbolic paraboloid, prove that there exist exactly 2 lines that pass through that point and lie on the surface of that paraboloid.

Kendrick Pierce 2022-05-21

You can use the dot product to determine if two vectors are parallel by seeing if their dot product is = 1. What about if they are parallel but point in the opposite direction? Would the dot product just equal -1 in this case?

Aditya Erickson 2022-05-21

I got a plane with equation: $P (x, y) = \frac{16100 + 47 y}{52}$
A line (parametrized equation) which lies on the plane P:
${\begin{matrix} x = t \\ y = 620 \\ z = 870 \end{matrix}$
And a point which belongs to the line: $O (- 5, 620, 870)$
I need to find an equation of the line that belongs to the specified plane P and crosses specified line at the specified point O making an angle of $45^{o}$ with the line.

Brooke Webb 2022-05-21

$K \in R_{+}$ and $z \in [0, x] \cap [0, K]$ $γ > 0$ one defines
$\begin{aligned} ϕ (z) = (K - z) {(\frac{z}{x})}^{γ} \end{aligned}$
The maximum is attained at $z = \frac{γ K}{γ + 1}$ . Question: Why is for $x \leq \frac{γ K}{γ + 1}$ :
$\begin{aligned} max_{z} ϕ (z) = K - x \end{aligned}$
And for $x > \frac{γ K}{γ + 1}$
$\begin{aligned} max_{z} ϕ (z) = ϕ (\frac{γ K}{γ + 1}) ? \end{aligned}$
Best regards.

America Ware 2022-05-21

I would like to maximize the Conﬂuent Hypergeometric Distribution in order to apply a Rejection sampling. The formula of the distribution is
$f (x; a, b, c) = K x^{a - 1} (1 - x)^{b - 1} e^{- c x}$
where $0 \leq x \leq 1$ and $a, b > 0, c \in R$
and let us assume that K is constant in terms of x. I would like to maximize this distribution in order to find an upper bound C and apply the rejection sampling.

Ethan Lee2022-05-20

Find the surface area of the prism.

Geometry help for high school students is one of the most popular requests online as Geometry always requires examples to learn. It is one of the reasons why high school geometry examples that have been provided will help you find the answers to the most common questions, be it graphs or the use of algebraic equations. If you require help with high school geometry, compare your instructions with the questions that have been shared below. The same relates to high school geometry test questions that are provided with multiple-option examples. Dealing with analytical geometry problems, always focus on word challenges.

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