Proof Involving the Diagonals of Parallelogram

[nycmathguy]

Homework Statement

Algebra & Geometry

Relevant Equations

Geometric Proof Using Algebra

Chapter 1, Section 1.1.
Look at the picture. Question 57.

Let me see.

To show this prove, I must find the midpoint of the diagonals. The midpoint of (b, c) and (a, 0) must be the same as the midpoint of (0, 0) and
(a + b, c).

You say?

 

[Ssnow]

I think it is sufficient to see in the picture as you suggest. Analytically you can consider the equation of the diagonals, for example the equation of the diagonal between $(b,c)$ and $(a,0)$ is $y=\frac{c}{b-a}x-\frac{ca}{b-a}$. The diagonal between $(a+b,c)$ and $(0,0)$ is $y=\frac{c}{a+b}x$. Now you can try to solve the system of both finding the intersection point ...
Ssnow

 

[nycmathguy]

I think it is sufficient to see in the picture as you suggest. Analytically you can consider the equation of the diagonals, for example the equation of the diagonal between $(b,c)$ and $(a,0)$ is $y=\frac{c}{b-a}x-\frac{ca}{b-a}$. The diagonal between $(a+b,c)$ and $(0,0)$ is $y=\frac{c}{a+b}x$. Now you can try to solve the system of both finding the intersection point ...
Ssnow

How did you come up with the two equations given in the information in the diagram?

 

[Mark44]

Homework Statement:: Algebra & Geometry

This is not a homework statement.
An informative homework statement would be "Show or prove that the diagonals of a parallelogram intersect at their midpoints."

Relevant Equations:: Geometric Proof Using Algebra

Not an equation and not relevant. Relevant equations would include those for the midpoint of a line segment and possibly the equation of a line.

Chapter 1, Section 1.1.

Not useful if we don't know which textbook you're using.

Look at the picture. Question 57.

Let me see.

Stream of thought fluff like "let me see" is not useful. It's better to leave this out, especially since you have done this in several of your threads.

To show this prove, I must find the midpoint of the diagonals. The midpoint of (b, c) and (a, 0) must be the same as the midpoint of (0, 0) and
(a + b, c).

Right, and here is where you could have put your calculations.

You say?View attachment 284440

 

[Mark44]

How did you come up with the two equations given in the information in the diagram?

He did it using an equation for a line, of which there are at least three. This would have been useful to include in the "Relevant Equations" section of the homework template.

The picture gives the coordinates of the endpoints of the diagonals, so it's just a matter of plugging these values in.

Here are three different equations for a straight line:

• Slope-intercept form: $y = mx + b$ -- Use it if you know the slope of the line and its y-intercept.
• Point-slope form: $y - y_0 = m(x - x_0)$ -- Use it if you know the slope of the line and a point on the line. This is the equation form that @Ssnow used.
• Standard form: $ax + by + c = 0$ -- Every line, including vertical lines, can be written using this equation form.

 

[nycmathguy]

He did it using an equation for a line, of which there are at least three. This would have been useful to include in the "Relevant Equations" section of the homework template.

The picture gives the coordinates of the endpoints of the diagonals, so it's just a matter of plugging these values in.

Here are three different equations for a straight line:

• Slope-intercept form: $y = mx + b$ -- Use it if you know the slope of the line and its y-intercept.
• Point-slope form: $y - y_0 = m(x - x_0)$ -- Use it if you know the slope of the line and a point on the line. This is the equation form that @Ssnow used.
• Standard form: $ax + by + c = 0$ -- Every line, including vertical lines, can be written using this equation form.

I know how to use those equations for simple questions not for anything like this thread.

 

[Mark44]

I know how to use those equations for simple questions not for anything like this thread.

Since your goal is to study calculus, that's why I suggested a review of precalc material. Any calculus book will assume that you understand and know how to use equations of a line, as well as a lot of other precalc topics.

If you're having trouble with the problem in this thread, that's an even stronger argument for spending time on precalculus stuff.

Despite all of my criticisms, you're doing a lot of things right, and should be commended for your goal of studying calculus. A lot of what I'm telling you is my attempt to help you get your thoughts organized so you can be successful in your goal. I'm also trying to get you to stretch out a bit and risk making mistakes.

 

[nycmathguy]

Since your goal is to study calculus, that's why I suggested a review of precalc material. Any calculus book will assume that you understand and know how to use equations of a line, as well as a lot of other precalc topics.

If you're having trouble with the problem in this thread, that's an even stronger argument for spending time on precalculus stuff.

Despite all of my criticisms, you're doing a lot of things right, and should be commended for your goal of studying calculus. A lot of what I'm telling you is my attempt to help you get your thoughts organized so you can be successful in your goal. I'm also trying to get you to stretch out a bit and risk making mistakes.

I started precalculus again back to chapter 1 or have you not noticed my precalculus questions already posted? I am going to tackle precalculus and calculus 1 at the same time but at a slower pace. I am also going to reduce the number of questions posted per week.

 

[Delta2]

I am also going to reduce the number of questions posted per week.

No need to do that, but if you feel confident about a solution to a problem just don't post it. Some moderators say that you are not doing a good use of PF if you post easy (for them) problems.

 

[Mark44]

I know how to use those equations for simple questions not for anything like this thread.

I don't understand why this is difficult for you. You're given the coordinates of all of the points, so as I said before, it's straightforward to plug them into one of the formulas for the equation of a line. Is the problem that the coordinates are given in variables like a and b rather than specified constants like 2 and 5?

I started precalculus again back to chapter 1 or have you not noticed my precalculus questions already posted?

Of course I noticed -- I've responded to most of your threads.

 

[Delta2]

Maybe what @Mark44 forgot to tell you in his nice post #5 is that the slope $m$ of a line that connects points (a,b) and (c,d) is $m=\frac{d-b}{c-a}$

 

[nycmathguy]

I don't understand why this is difficult for you. You're given the coordinates of all of the points, so as I said before, it's straightforward to plug them into one of the formulas for the equation of a line. Is the problem that the coordinates are given in variables like a and b rather than specified constants like 2 and 5?

Of course I noticed -- I've responded to most of your threads.

It is not difficult to solve. It is just difficult to set up or better yet, get started.

 

[Mark44]

It is not difficult to solve. It is just difficult to set up or better yet, get started.

This doesn't make sense. If one can't figure out how to set up a problem, there's no chance of solving it.

 

[nycmathguy]

This doesn't make sense. If one can't figure out how to set up a problem, there's no chance of solving it.

I am talking mainly about applications. There are certain word problems (system of equations, for example) that requires setting up equations in two and sometimes three unknowns. My thread about finding two missing numbers is a perfect example.