the quadrilateral

[logistic_guy]

here is the question




my attemb

\(\displaystyle XW + WV + VT + TX = 8.3 + 3.1 + 3.3 + 4.5 + 1.2 = 20.4\)

it say wrong

 

[Dr.Peterson]

here is the question

View attachment 38870


my attemb

\(\displaystyle XW + WV + VT + TX = 8.3 + 3.1 + 3.3 + 4.5 + 1.2 = 20.4\)

it say wrong

That's not a question, it's a picture!

What are you trying to do with it?? Were there any words in the problem?

If you want the perimeter, then of course it's correct -- unless (as I strongly suspect) the 1.2 does not represent all of TX. The picture isn't very carefully labeled (or, rather, explained).

Do you see how to find the other part of TX? And do you see that the 1.2 could have been found from the other numbers? (Hint: the two tangent segments from a given point are equal.)

 

[logistic_guy]

That's not a question, it's a picture!

how the question isn't therei'm sure i write

here is the question

Find the perimeter of the quadrilateral.

If you want the perimeter, then of course it's correct -- unless (as I strongly suspect) the 1.2 does not represent all of TX. The picture isn't very carefully labeled (or, rather, explained).

if the drawing isn't clear i can draw big one

Do you see how to find the other part of TX? And do you see that the 1.2 could have been found from the other numbers? (Hint: the two tangent segments from a given point are equal.)

i see it now i think this what you mean \(\displaystyle 4.5 - 3.3 = 1.2\) so \(\displaystyle 8.3 - 3.1 = 5.2\)

\(\displaystyle XW + WV + VT + TX = 8.3 + 3.1 + 3.3 + 4.5 + 1.2 + 5.2 = 25.6\)
this give correct answer

(Hint: the two tangent segments from a given point are equal.)

i don't know this before. how to proof?


thank Dr. i appreciate your quick help

 

[Dr.Peterson]

Yes, you've solved it.

Here is one source for the fact you needed to use:

7.3: Tangents to the Circle

A tangent to a circle is a line which intersects the circle in exactly one point.

math.libretexts.org

(See theorem 7.3.3 and click on "Proof".)

 

[logistic_guy]

Yes, you've solved it.

Here is one source for the fact you needed to use:

7.3: Tangents to the Circle

A tangent to a circle is a line which intersects the circle in exactly one point.

math.libretexts.org

(See theorem 7.3.3 and click on "Proof".)

thank Dr. very much

theorem 7.3.3 depnd on theorem 7.3.1

this theeorem i don't like. it's not convincing me or maybe it's written informally?

 

[Dr.Peterson]

Is it the theorem, or the proof, that you don't like? (And do you mean 7.3.3 or 7.3.1?) What part of the proof is not convincing? (Not that I was recommending this as the best textbook ever, just to show that it's a well-known fact. And you can find other proofs of either theorem easily enough.)

I notice that in the proof of 7.3.1 they mention "Theorem 7.3.2, Section 4.6", when they mean Theorem 4.6.2. So there are typos there.

 

[logistic_guy]

Is it the theorem, or the proof, that you don't like? (And do you mean 7.3.3 or 7.3.1?) What part of the proof is not convincing? (Not that I was recommending this as the best textbook ever, just to show that it's a well-known fact. And you can find other proofs of either theorem easily enough.)

I notice that in the proof of 7.3.1 they mention "Theorem 7.3.2, Section 4.6", when they mean Theorem 4.6.2. So there are typos there.

the proof of theorem 7.3.1 isn't convencing.

it say the shortest line segment that can be drawn from a point to a straight line is the perpendicular. i can draw three tangent lines on the same point on circle, non of them is perpendicular to the radius.

i'm sure you study calculus very much. if the angle between two intersecting is > ninty degree by a very small amount, calculus will disagree with me, but i consider the lines not perpendicular even if the angle is only \(\displaystyle 90.0000000000001\)

if the theorem is infomrmal, i agree with it. it should use a limit to convince me more

 

[Dr.Peterson]

the proof of theorem 7.3.1 isn't convincing.

Here's the proof of 7.3.1, so we have it in front of us:

​

it say the shortest line segment that can be drawn from a point to a straight line is the perpendicular.

It is not Theorem 7.3.1 that says this; it's 4.6.2. Are you saying you don't believe this statement, or just that you don't think it applies in the proof of 7.3.1? If the former, did you look at its proof?

i can draw three tangent lines on the same point on circle, non of them is perpendicular to the radius.

What? Show me what that looks like. Surely you're misinterpreting something, such as the definition of "tangent".

i'm sure you study calculus very much. if the angle between two intersecting is > ninety degree by a very small amount, calculus will disagree with me, but i consider the lines not perpendicular even if the angle is only 90.0000000000001.

That's correct, even in calculus: equality must be exact to be meaningful. Any tangent line is exactly perpendicular to the radius at that point.

if the theorem is informal, i agree with it. it should use a limit to convince me more

Why would that be more convincing? Limits are harder to fully understand than simple geometrical facts, and lead to many misunderstandings.

 

[logistic_guy]

Here's the proof of 7.3.1, so we have it in front of us:

View attachment 38876​

It is not Theorem 7.3.1 that says this; it's 4.6.2. Are you saying you don't believe this statement, or just that you don't think it applies in the proof of 7.3.1? If the former, did you look at its proof?


What? Show me what that looks like. Surely you're misinterpreting something, such as the definition of "tangent".


That's correct, even in calculus: equality must be exact to be meaningful. Any tangent line is exactly perpendicular to the radius at that point.

Why would that be more convincing? Limits are harder to fully understand than simple geometrical facts, and lead to many misunderstandings.

it don't matter theorem 7.31 or theorem 4.6.2 say the perpedicular thing

my point i'm say this



i can draw million line tangent to circle at point [imath]P[/imath] and they all not perpednicular

Why would that be more convincing? Limits are harder to fully understand than simple geometrical facts, and lead to many misunderstandings.

because the limit will make sure the angle go to \(\displaystyle 90\)

the thereeom must say one of these line is perpednicular to the radius at point \(\displaystyle P\) when \(\displaystyle \lim_{\theta \to 90^{\circ}}\theta\). this angel is between the radius and the line in either side

as you can see all of the three line are tangent to circle at point \(\displaystyle P\), but non of them is perpedicular.

 

[Dr.Peterson]

i can draw million line tangent to circle at point P and they all not perpednicular

How do you define tangent?

Whether you're thinking in terms of geometry or calculus, there is no way you could make this claim. A circle has exactly one tangent line at a given point, and any differentiable curve has only one tangent line at a given point.

 

[Aion]

As was stated above a common definition is the following:

A straight line that has exactly one point in common with a circle is called a tangent to the circle.

By this definition each tangent to a circle at a given point is unique. This is because a tangent to a circle is defined as a line that touches the circle at exactly one point. At this point of tangency, the tangent is perpendicular to the radius drawn to that point (theorem). Since there can only be one unique line perpendicular to a radius at a given point on the circle, there can only be one unique tangent at that point.

 

[logistic_guy]

How do you define tangent?

Whether you're thinking in terms of geometry or calculus, there is no way you could make this claim. A circle has exactly one tangent line at a given point, and any differentiable curve has only one tangent line at a given point.

i don't know the definetion. i told you before calculus won't agree with me. if definetion and theorem and calculus say it tangent to one point only. why i can draw different tangent at same point in my post 9? contadiction?

As was stated above a common definition is the following:

A straight line that has exactly one point in common with a circle is called a tangent to the circle.

By this definition each tangent to a circle at a given point is unique. This is because a tangent to a circle is defined as a line that touches the circle at exactly one point. At this point of tangency, the tangent is perpendicular to the radius drawn to that point (theorem). Since there can only be one unique line perpendicular to a radius at a given point on the circle, there can only be one unique tangent at that point.

thank Aion

you explaine it nicely. appreciate your help

 

[Dr.Peterson]

why i can draw different tangent at same point in my post 9? contadiction?

Those are not tangents. Please tell us why in the world you think they are.