Working with Perpendicular Segments

[nycmathguy]

Homework Statement

Use the distance formula and the Pythagorean theorem in terms of working with perpendicular segments.

Relevant Equations

$distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

(leg)^2 + (leg)^2 = (hypotenuse)^2

For easy calculation, I will use a for m_1 and b for m_2 and then back substitute for a and b.

We have (0, 0) and (1, a).

d_1 = sqrt{(1 - 0)^2 + (a - 0)}

d_1 = sqrt{(1)^2 + (a)^2}

d_1 sqrt{1 + a^2}

For d_2, we are going to need (0, 0) and (1, b).

I say d_2 = sqrt{1 + b^2}.

Back-substitute for a and b.

d_1 = sqrt{1 + m_1}

d_2 = sqrt{1 + m_2}

To find the distance from (1, m_1) to (1, m_2), I can use the distance formula or the Pythagorean Theorem.

I don't understand this part of the problem:

"Then use the Pythagorean Theorem to find a relationship m_1 and m_2."

Stuck here.

 

[BvU]

Use the distance formula and the Pythagorean Theorem in terms of working with perpendicular segments.

Nice suggestion, but: what is the problem statement ?

$\$

 

[Delta2]

At start you effectively saying that a=m_1 and b=m_2 but then you actually saying that a^2=m_1 and b^2=m_2. You have to decide how exactly you relate a,m_1 and b , m_2.

Homework Statement:: Use the distance formula and the Pythagorean Theorem in terms of working with perpendicular segments.
Relevant Equations:: distance = sqrt{x_2 - x_1)^2 + (y_2 - y_1)^2}

(leg)^2 + (leg)^2 = (hypotenuse)^2

To find the distance from (1, m_1) to (1, m_2), I can use the distance formula or the Pythagorean Theorem.

You will do both ways and then equate them and thus you ll find an equation that relates m_1 and m_2.

 

[Delta2]

Nice suggestion, but: what is the problem statement ?

$\$

You are right he doesn't give the exact statement of the problem, I just figure it out on my own based on general context but I might be wrong.
The problem statement might be something like this: Find the relation between m_1 and m_2, such that the points (1,m_1) and (1,m_2) form perpendicular segments with the point (0,0).

 

[Mark44]

he doesn't give the exact statement of the problem, I just figure it out on my own based on general context but I might be wrong.

You shouldn't have to figure it out on your own. The Homework Statement should include the info of the problem.

 

[symbolipoint]

Homework Statement:: Use the distance formula and the Pythagorean Theorem in terms of working with perpendicular segments.
Relevant Equations:: distance = sqrt{x_2 - x_1)^2 + (y_2 - y_1)^2}

(leg)^2 + (leg)^2 = (hypotenuse)^2

I don't understand this part of the problem:

"Then use the Pythagorean Theorem to find a relationship m_1 and m_2."

The Pythagorean Theorem and the Distance Formula are really the same thing, but just in two different arrangements.-----
EDIT: Adding more to help in conceptual development

This is a way to find that yourself.

Take a typical high school Geometry book. Find the part which shows a proof for the Pythagorean Theorem.
Notice how the figure or diagram is arranged and labeled, and follow the written proof.
Now make a graph of two cartesian points; any two points. Use specific known points if you wish. Take points in the first quadrant to make the task easier. You should be able to find where the next point is IN ORDER TO FORM A RIGHT TRIANGLE.

Seeing your graph or diagram, you should easily identify the two legs and the hypotenuse. I am not explaining every detail in how to do this, but I say you need to do this on your own. I could create and label the graph and make a scanned file and write other information and upload to the topic for viewing on the forum, but I will not.

If you drew the graph well, you should have one vertical leg, one horizontal leg, and the triangle's hypotenuse.
Make the expression for length of one leg. Make the expression for the length of the other leg. Your hypotenuse is an unknown value.

USE the Pythagorean Theorem to write how the two legs and that hypotenuse are interrelated.

Now, solve for the hypotenuse. YOU NOW HAVE THE DISTANCE FORMULA.

 

[nycmathguy]

The Pythagorean Theorem and the Distance Formula are really the same thing, but just in two different arrangements.-----
EDIT: Adding more to help in conceptual development

This is a way to find that yourself.

Take a typical high school Geometry book. Find the part which shows a proof for the Pythagorean Theorem.
Notice how the figure or diagram is arranged and labeled, and follow the written proof.
Now make a graph of two cartesian points; any two points. Use specific known points if you wish. Take points in the first quadrant to make the task easier. You should be able to find where the next point is IN ORDER TO FORM A RIGHT TRIANGLE.

Seeing your graph or diagram, you should easily identify the two legs and the hypotenuse. I am not explaining every detail in how to do this, but I say you need to do this on your own. I could create and label the graph and make a scanned file and write other information and upload to the topic for viewing on the forum, but I will not.

If you drew the graph well, you should have one vertical leg, one horizontal leg, and the triangle's hypotenuse.
Make the expression for length of one leg. Make the expression for the length of the other leg. Your hypotenuse is an unknown value.

USE the Pythagorean Theorem to write how the two legs and that hypotenuse are interrelated.

Now, solve for the hypotenuse. YOU NOW HAVE THE DISTANCE FORMULA.

Let's not travel East, West, North and South. In short, what is the question asking for?

 

[Mark44]

Let's not travel East, West, North and South. In short, what is the question asking for?

Let us know what the question is -- you didn't include it, so we're just guessing what the question is.

 

[symbolipoint]

Let's not travel East, West, North and South. In short, what is the question asking for?

Let us know what the question is -- you didn't include it, so we're just guessing what the question is.

Not that. What I presented is a pathway to make a derivation. I did not express that as a question. I presented, 'do these things in this order, and you will have a derivation'. That long description was made in hope that the way the Distance Formula is related to Pythagorean Theorem may become very clear and understandable.

 

[Mark44]

Not that. What I presented is a pathway to make a derivation.

Apparently a derivation of the distance formula, but there was no actual statement of the problem, so it seems to me that your steps are just a guess at what the question is. In the OP there was mention of $m_1$ and $m_2$ which suggest to me that they represent slopes of two line segments.
Another interpretation of what might be asked here is to show the relationship between the slopes of perpendicular line segments.

@nycmathguy, unless you're headed off to FB and don't plan to return, it would be nice if you told us what the problem is, so we don't have to guess.

 

[Delta2]

Another interpretation of what might be asked here is to show the relationship between the slopes of perpendicular line segments.

Yes I believe that's what it is about.

To the OP:
Set $d^2=d_1^2+d_2^2$ where $d$ the distance between $(1,m_1)$ and $(1,m_2)$ and $d_1,d_2$ the distances of those points from (0,0). Then you ll be able to infer the relation between $m_1$ and $m_2$.

 

[symbolipoint]

Apparently a derivation of the distance formula, but there was no actual statement of the problem,-------. In the OP there was mention of $m_1$ and $m_2$ which suggest to me that they represent slopes of two line segments.
Another interpretation of what might be asked here is to show the relationship between the slopes of perpendicular line segments.

--- a part snipped out ---

He is trying to show points in the cartesian plane. He does not seem to be focused on slopes there and he does not need to be.

 

[Delta2]

its going to be worst at FB, just stay here is my recommendation.

 

[Mark44]

The OP has declined to tell us the exact problem statement, so I'm closing this thread. @nycmathguy, you can start a new thread with this question, provided that you give us the complete problem statement.