AI or GPT-4 try?

Can AI or GPT-4 handle this?
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Geometry problem: Triangle with known length sides a, b, c where a is the longest. Place inside a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths?

 

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"
We can start by drawing the triangle and the semicircle as described:

javascriptCopy code
c
+--------+
/ /
/ /
/ /
+--------+
b a

Since the diameter of the semicircle lies on side a, the radius of the semicircle is half the length of a. Let's call this radius R.

To find the largest possible value of R, we need to maximize the area of the semicircle. The area of a semicircle is given by:

A = (π/2) R^2

To maximize A, we need to maximize R^2. Let's use the Pythagorean theorem to find an expression for R^2 in terms of a, b, and c:

a^2 = b^2 + c^2

We can solve for c^2:

c^2 = a^2 - b^2

Substituting this into the formula for the area of the semicircle, we get:

A = (π/2) R^2 = (π/2) (a^2/4 - b^2/4)

Now we can maximize A by taking the derivative with respect to b and setting it equal to zero:

dA/db = -π/8 * b = 0

This implies that b = 0, which is clearly not the case. Therefore, the maximum value of A is obtained when b is as large as possible, which means b = a/2.

Substituting this value into the expression for A, we get:

A = (π/2) (a^2/4 - (a/2)^2/4) = (π/8) a^2

So the maximum area of the semicircle is (π/8) a^2, which means the radius of the largest possible semicircle is:

R = a/2 * sqrt(π/2)

Therefore, the radius of the largest possible semicircle in terms of side lengths is:

R = a/2 * sqrt(π/2)"
From ChatGPT 3.5

 

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I must be misunderstanding the parameters. Why would the solution not be a right triangle, where b and c have length 2 root a, and the semicircle has radius a/2?

I guess this needs a diagram.

 

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"You are correct that if we choose b = c = 2√a and a = a, then the semicircle with diameter on side a will have radius a/2, and this triangle satisfies the conditions given in the problem.

However, it is not necessarily the case that this is the only solution. There may be other triangles that also satisfy the conditions of the problem and have a larger semicircle. In fact, we have shown that for any given triangle with sides a, b, and c satisfying a^2 = b^2 + c^2, the radius of the largest possible semicircle with diameter on side a is a/2 * sqrt(π/2).

Therefore, while the triangle you described is one possible solution, it is not necessarily the only solution."

 

errors: radius is NOT half length of a, triangle is any with a largest.

 

Why do you say this?
The problem statement says "semi-circle with diameter on side a". There is nothing in there that says the semicircle's diameter cannot be 2a.

 

It does not say it is either, so why make that assumption? Moreover semi-circle INSIDE triangle makes it impossible for diameter to be side a.

 

It's not an assumption; the problem asks for the largest semi-circle possible. The semi circle will be maximized if it has diameter a.

But that's predicated on the following:

The triangle is inside the semicircle, to-wit:

"[A] triangle... Place (it) inside a semi-circle with diameter on side a."


See, I questioned the wording of the text, wondering if, perhaps, it was meant to be

"[A] triangle... Place (the following) inside: a semi-circle with diameter on side a."

which would be interpreted as you suggest.


The problem is: if read the way you interpret it has to be grammatically incorrect (missing punctuation), whereas the way I interpret it, it is not incorrect, only awkward and ambiguous.

And this is why I insist that it needs a diagram. The problem statement - as you have written it - is demonstrably ambiguous.

 

Corrected statement
Geometry problem: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths?

 

OK, this:

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This seems a long-winded way of describing the setup.

The largest possible interior of the triangle while maintaining a > b and a > c is when b and c are both infinitesimally shorter than a. In other words, a verrrrry slightly squashed equilateral triangle.

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So, the question reduces to: what is the largest semicircle that can fit inside an equilateral triangle (without the curve actually touching it)?

 

My intention: Sides b and c are arbitrary fixed in advance, NOT subject to change.

 

Why don't you ask a GPT3 or 4 exactly as you stated above.? Ask a GPT that is mathematically trained.
I bet it will have an answer in about 3 seconds.

It is these mathematical problems that computers do! Althought GPT is not specifically designed for solving complex mathematical problems, it can learn and even learn by itself with some instruction.

Meet MathGPT: a Chatbot Tutor Built Specific to a Math Textbook

• By Kristal Kuykendall
• 01/30/23

more.... https://thejournal.com/articles/202...-tutor-built-specific-to-a-math-textbook.aspx

 

Following statement posed to GPT - no answer!

Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with diameter on side a. What is radius of largest possible semi-circle in terms of side lengths? Position of diameter center along a?

 

I am not surprised. How long does it take a human to learn such advanced maths, 15 years?
The new AIs are not specialized number crunchers. They are text-based minds just like humans.

That is why I said the new AI needs to be trained (in school) just like people. Or if you will, you can teach the AI to do its own research, which is more prone to flaws, just as any individual human can study and still not quite get it right if the source is suspect.

Theoretical mathematics is a science and requires in-depth "knowledge" of mathematics.

 

"
This is a challenging problem. I will try to help you by using some facts from geometry.

12, which says that if a triangle is inscribed inside a circle, where one side of the triangle is the diameter of the circle, then the angle opposite to that side is a right angle.

So, if we draw a semi-circle with diameter on side a, then the angle opposite to a is a right angle. Let’s call this angle C.

Now, we can use the Pythagorean theorem to find the length of a in terms of b and c. We have:

a2=b2+c2

3 to find the radius of the semi-circle in terms of a and C. We have:

fracasinC=2R

where R is the radius of the semi-circle.

Since we know that C is a right angle, we have:

sinC=1

So, we get:

a=2R

Substituting this into the Pythagorean theorem, we get:

(2R)2=b2+c2

Solving for R, we get:

R=fracsqrtb2+c22

This is the radius of the largest possible semi-circle in terms of side lengths.

To find the position of the diameter center along a, we can use similar triangles. We have:

fracRb=fraca−Rc

Substituting a = 2R, we get:

fracRb=fracRc

Cross-multiplying, we get:

cR=bR

Dividing by R, we get:

c=b

This means that the triangle is an isosceles right triangle, and the diameter center is at the midpoint of a.

I hope this helps you with your problem. "
From Bing Chat

 

I'd say that's a string indication that the question is poorly formed.

 

Except that is not the problem as stated. You've gotten it inside-out.
The problem as stated is a semicircle inside a triangle.

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I'm just giving you the Bing Chat answer.

 

I don't know if anybody noticed the author of that equation,
i.e. ChatGPT 3.5

Perhaps the question was not posed correctly, but the answer is nevertheless impressive.

 

Geometry problem: Semi-circle inside triangle: Triangle with known length sides a, b, c where a is the longest. Place inside the triangle a semi-circle with entire diameter resting on side a, which is horizontal. The arc of the semicircle is maximum possible inside the triangle.
Known - triangle dimensions. Unknown r (semi-circle arc radius) and x (distance along a from left end of a to center of diameter). Find equations for r and x In terms of all possible triangles.
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Above is my latest attempt ro check AI. It was submitted to both BingChat and Chatgpt. Neither got it right.