High School Geometry Examples and Practice Questions

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High school geometry

Recent questions in High school geometry

trumansoftjf0 2022-11-11

Computing the sides of inscribed polygons
For regular polygons inscribed in a circle of radius 1, use $S (6) = 1$ to conclude that:
$S (12) = \sqrt{2 - \sqrt{3}}$
$S (24) = \sqrt{2 - \sqrt{2 + \sqrt{3}}}$

Clara Dennis 2022-11-10

Distance between (-7,6,10) and (7,-4,9)?

figoveck38 2022-11-10

Understanding shell height (finding volume)
- Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under curve $y = \sqrt{x}$ from 0 to 1.
Shell radius $= y$
Shell height $= 1 - y^{2}$
$V = 2 π \int_{0}^{1} y (1 - y^{2}) d y$
I didn't understand the shell height, why it is $1 - y^{2}$ ? Why not $y^{2}$ ? I think, shell height should be $y^{2}$ . Can you explain?

nyle2k8431 2022-11-10

Let E be the event of an even number of successes.
$u_{n}$ : Probability of E occurring at the nth trial not necessarily for the first time
$f_{n}$ : Probability of E occurring at the nth trial for the first time
Let U(x) and F(x) be the corresponding probability generating functions and from that we have the equation
$U (x) = 1 / (1 - F (x))$

Juan Lowe 2022-11-10

Distance between (2,7,4) and (7,4,1)?

Rosemary Chase 2022-11-10

The volume of the region between two spheres and the upper nappe of a cone
I am really having trouble constraining the region between these three surfaces. I am imagining a sort of "Dome", or a "muffin head" sort of shape. Is this correct ? Anyway, I need to be able to write the following volume integral in rectangular, cylindrical and spherical coordinates:
Consider the region that is between $x^{2} + y^{2} + z^{2} = 1$ , $x^{2} + y^{2} + z^{2} = 9$ , and finally above the upper nappe of the cone $z^{2} = 3 (x^{2} + y^{2})$ .
Upon further consideration, does the smaller sphere even matter?

undergoe8m 2022-11-09

How come if X is a Geometric random variable, then $P (X > x) = q^{x} ?$

Messiah Sutton 2022-11-09

Volume by disk perpendicular to x axis
Find the volume of the solid formed when region enclosed by $y = x^{1 / 2}, y = 6 - x$ , $y = x^{1 / 2}, y = 6 - x$ and $y = 0$ revolves around x axis.
Can I use the equation for finding solids by splitting the solid into 2 parts?

kemecryncqe9 2022-11-09

What is the angle sum of a polygon with 9 sides?

inurbandojoa 2022-11-08

Probability that at least 4 out of 365 times I try, I need more than 5 flips before getting a tails
Suppose I flip a coin until I get a tails. Let X be the number of flips this takes. What is the probability that there are at least 4 days in a year where I needed more than 5 flips to get a tails?
Is this kind of question combining geometric and exponential distributions with some kind of conditional probability? I'm unsure how to compute this.

perlejatyh8 2022-11-08

Expected area of a random triangle with fixed perimeter
I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1.
My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the varying areas. But then I realized that I wouldn't account for ALL triangles using that method. For example, an equilateral triangle with side lengths one third would not be included.

reevelingw97 2022-11-08

Bounded area for any triangle formed by polygons
Let $P_{1}, P_{2}, P_{3}$ be closed polygons on the plane. Suppose that for any points $A \in P_{1}$ (meaning A can be inside or on the boudary of $P_{1}$ ), $B \in P_{2}, C \in P_{3}$ , we have $[A B C] \leq 1$ . Is it possible that two of $P_{1}, P_{2}, P_{3}$ have area $\geq 4$ ? What about all three having area $\geq 4$ ?
Here, [X] denotes the area of the polygon X.

Hanna Webster 2022-11-08

The length of a rectangle is 14 m longer than its width and the area of the rectangle is 400 m2. Find the exact dimensions of the rectangle: width and length.

Emmanuel Giles 2022-11-08

An identity involving geometric distribution
Let $(Ω, A, P)$ be probability space and $X : Ω \to N_{0}$ be a geometrically distributed random variable with parameter $p \in (0, 1)$ $X ∽ Geo (p)$ . Show that $P (X = m + n | X \geq m) = P (X = n), for all m, n \in N_{0} .$

assupecoitteem81 2022-11-08

Finding the volume of a solid by rotating two curves about the y-axis
$y = 56 x - 7 x^{2} and y = 0$
My issue with this question is that I am having trouble turning the equation, $y = 56 x - 7 x^{2}$ , in terms of y. I understand that by doing that, I can proceed with the integration... Perhaps there is another method to do this without having to turn it in terms of y?

Uriel Hartman 2022-11-08

Number of r-sided polygons in P with no common edges
We have a n-sided convex polygon P. How many r-sided polygons $(r < n)$ , with its vertices among those of P, can be formed such that it has no sides (edges) in common with P?
I tried to use the inclusion-exclusion principle, defining $A_{m}$ to be the property that our polygon has at least m sides in common with P. Then we need to find $(\binom{n}{r}) - | A_{1} \cup A_{2} \cup \dots \cup A_{r - 1} |$ . But I didn't make much progress along these lines, as finding $| A_{m} |$ for $m > 2$ is not easy.

Kareem Mejia 2022-11-08

Every polygon with an even number of vertices may be partitioned by diagonals into quadrilaterals.
True or False?

Laila Murphy 2022-11-08

In a regular polygon each interior angle is $135^{\circ}$ greater than each exterior angle. How many sides has the polygon?

inurbandojoa 2022-11-08

An upper bound on the expected value of the square of random variable dominated by a geometric random variable
Let X and Y be two random variables such that:
1. $0 \leq X \leq Y$
2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).
I would be grateful for any help of how one could upperbound $E (X^{2})$ in terms of p.

Keshawn Moran 2022-11-07

Direction in 3d space described with 2 values: Horizontal and Vertical (x and y) ranging the full 360 degrees (from -180 to +180), thus describing every possible view direction.
It seems fairly straight forward to me, that the angle between (0,0) and (5,0) would be 5 degrees.
But what is the angle between (0,0) and (5,5). If it was 2 points in 2d I would use Pythagoras theorem; having the difference be $\sqrt{(} 5^{2} + 5^{2})$ . But I don't think that would apply in 3d? How do I go about calculating the angle between these 2 directions?

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