Recent questions in Elementary geometry

Properties of parallelograms
Answered

Brunton39
2022-06-23

$det(A)<0$

I knew about some of determinant of matrix properties as following, but it seems to me that it is nothing relative to negative value of determinant of matrix

$det(AB)=det(A)det(B)(Multiplicative)$

$det(A)=\mathit{\text{0}}\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}\mathit{\text{A is singular}}$

${M}_{2,2}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$|det({M}_{2\times 2})|=|ad-bd|=\mathit{\text{volumn of parallelogram}}$

$|det({M}_{n\times n})|=\prod _{j=1}^{n}{a}_{i,j}(-1{)}^{i+j}det({M}_{i,j})\phantom{\rule{1em}{0ex}}\mathit{\text{expansion of determinant alone the}}{\mathit{\text{i}}}^{th}\mathit{\text{row}}$

Euclidean Geometry
Answered

Armeninilu
2022-06-23

Angle Bisectors
Answered

telegrafyx
2022-06-22

Well, $AC<AB$, but is there a formula for angle bisector length?

Midpoint Formula
Answered

Cory Patrick
2022-06-22

Euclidean Geometry
Answered

Devin Anderson
2022-06-22

Euclidean Geometry
Answered

Kyla Ayers
2022-06-22

Midpoint Formula
Answered

Jaqueline Kirby
2022-06-22

Angle Bisectors
Answered

Finley Mckinney
2022-06-21

I am unsure of how to find the equations. Any help will be greatly appreciated!

Midpoint Formula
Answered

Devin Anderson
2022-06-21

Postulates
Answered

Jaqueline Kirby
2022-06-21

1. Each pair of points can be joined by one and only one straight line segment.

2. Any straight line segment can be indefinitely extended in either direction.

3. There is exactly one circle of any given radius with any given center.

4. All right angles are congruent to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:

1. These to me sounds more like something that shouldn't require proving... does it?

2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?

3. Similarly, what space can allow two circle of the same radius and center to be not the same?

4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?

5. Why do we need the 5th postulate?

Angle Bisectors
Answered

Sarai Davenport
2022-06-20

I solved this without using vectors to get some idea. I am not sure how to prove it using vectors. I don't want to use vector equations for straight lines and then find the point of concurrency. That's like solving using coordinate geometry.

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ represent the sides $A,B,C$ respectively.

The angle bisectors are along $\frac{\overrightarrow{a}}{|\overrightarrow{a}|}}+{\displaystyle \frac{\overrightarrow{b}}{|\overrightarrow{b}|}},\phantom{\rule{1em}{0ex}}{\displaystyle \frac{\overrightarrow{b}}{|\overrightarrow{b}|}}+{\displaystyle \frac{\overrightarrow{c}}{|\overrightarrow{c}|}},\phantom{\rule{1em}{0ex}}{\displaystyle \frac{\overrightarrow{c}}{|\overrightarrow{c}|}}+{\displaystyle \frac{\overrightarrow{a}}{|\overrightarrow{a}|}$

Let the sides $AB,BC,CA$ be $x,y,z$. Let $AD$ be one of the angular bisector.

$\frac{BD}{CD}=\frac{x}{z}$

Hence

$D=\frac{x\overrightarrow{c}+z\overrightarrow{b}}{x+z}$

What should be the next step? Or is there a better method?

Euclidean Geometry
Answered

Quintin Stafford
2022-06-20

Midpoint Formula
Answered

April Bush
2022-06-18

Postulates
Answered

Theresa Archer
2022-06-18

As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.

As a good starting point, I'd like to better understand what the difference is between an axiom, a theorem and a postulate. At my current level of knowledge i would use them interchangeably (lol), however I'm sure one is founded upon the others.

If someone could explain the logical hierarchy/relation between these three it would be greatly appreciated.

Properties of parallelograms
Answered

arridsd9
2022-06-17

-an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality

-a norm can be induced by a metric if and only if the metric satisfies $d(x+a,y+a)=d(x,y)$ and $d(ax,ay)=ad(x,y)$

or are the implications one way?

Properties of parallelograms
Answered

Kendrick Hampton
2022-06-17

I'm defining my rhombus as follows: $[(0,0),(a,0),(b,c),(a+b,c)]$

I've managed to figure out that $c=\sqrt{{a}^{2}-{b}^{2}}$ and that the slopes of the diagonals are $\frac{\sqrt{{a}^{2}-{b}^{2}}}{a+b}$ and $\frac{-\sqrt{{a}^{2}-{b}^{2}}}{a-b}$

What I can't figure out is how they can be negative reciprocals of one another.

I mean to say that I could not find the algebraic proof. I've seen and understand the geometric proof, but I needed help translating it into coordinate form.

Midpoint Formula
Answered

minwaardekn
2022-06-16

$d(XY)=\frac{1}{2}ln{\textstyle (}\frac{\overline{XQ}\cdot \overline{YP}}{\overline{XP}\cdot \overline{YQ}}{\textstyle )}$

where P and Q are ideal points lying on the boundary of the unit disc, and $\overline{XQ}$ denotes the standard Euclidean distance between a point X inside the unit disc and an ideal point Q.

Given the ideal points $P=(0,1),Q=(0,-1)$ , nad points $A=(0,0)$ and $B=(0,\frac{1}{2})$ . I am asked to find the midpoint $M=(0,m)$ between A and B.

Properties of parallelograms
Answered

Feinsn
2022-06-16

The space c of all convergent sequences is also complete, as a closed subspace of ${l}^{\mathrm{\infty}}$, and so is ${c}_{0}$ - the space off all sequences with entries equal to zero from some point on.

I also know that both $c$ and ${c}_{0}$ are separable.

My question is - are $c$ and ${c}_{0}$ inner product spaces. Should I look for an example of sequences which don't satisfy the parallelogram law.

While elementary geometry is often encountered by college students as they are dealing with the basic design and modeling tasks, it is often necessary to implement equations or turn to Euclidean Geometry to determine the values of shapes and figures that are related to axioms and different theorems. When you are dealing with flat surfaces, it will help you to work with straight line segments. The same relates to those cases when you need to calculate formulas by using midpoint formula example problems that you can find below as you are dealing with the elementary problems in Geometry. If you find it ov