What is the value of b/a ?

@thebigk • Oct 22, 2024

Oct 22, 2024

1.3K

Here is yet another puzzle from Archimedes Lab:

[FONT=Verdana, Arial, Helvetica, sans-serif]Two intersected semi-circles inscribed in a rectangle are tangent to 5 discs (soccer balls) as depicted in the image. What is the value of b/a?

[IMG]

Puzzle taken from #-Link-Snipped-#

Get all mathy now! Happy Brainache 😁😁

-The Big K-
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Rocker

@rocker-ZpZLKM • Apr 24, 2007

Looking at the diagram, my best guess is '1'. Is that the right answer??
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Prasad Ajinkya

@prasad-aSUfhP • Apr 24, 2007

I think the answer is sqrt(2)

Construction: Let the centers of the two big semicircles be A and B. Let the semicircles cut each other at C and D. So, we have a quadrilateral ABCD.

Proof: Since AC and AD are tangents to circle B, and BC and BD are tangents to circle A. We can conclude that ACB and BDA are right angles. Also, since all of the sides of this quadrilateral are radii or either circle A or circle B, which equal to a length of b/2. Thus, ABCD is a square of sides b/2. Diagonal AB is of the length sqrt(2)*b/2 = b/sqrt(2).

Also, len(AB) = a = b/sqrt(2). Thus, b/a = sqrt(2). Ta-da!
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Rocker

@rocker-ZpZLKM • Apr 25, 2007

kidakaka
I think the answer is sqrt(2)

Construction: Let the centers of the two big semicircles be A and B. Let the semicircles cut each other at C and D. So, we have a quadrilateral ABCD.

Proof: Since AC and AD are tangents to circle B, and BC and BD are tangents to circle A. We can conclude that ACB and BDA are right angles. Also, since all of the sides of this quadrilateral are radii or either circle A or circle B, which equal to a length of b/2. Thus, ABCD is a square of sides b/2. Diagonal AB is of the length sqrt(2)*b/2 = b/sqrt(2).

Also, len(AB) = a = b/sqrt(2). Thus, b/a = sqrt(2). Ta-da!
well done kidakaka😁! I think you got the correct answer!😁😁😁
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Kaustubh Katdare

@thebigk • Apr 25, 2007

Right or Wrong, we'll have to wait for the right answer 😁!

Others, what's your answer?

-The Big K-
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NiLaY

@nilay-lpoAmu • Apr 26, 2007

kidakaka
I think the answer is sqrt(2)

Construction: Let the centers of the two big semicircles be A and B. Let the semicircles cut each other at C and D. So, we have a quadrilateral ABCD.

Proof: Since AC and AD are tangents to circle B, and BC and BD are tangents to circle A. We can conclude that ACB and BDA are right angles. Also, since all of the sides of this quadrilateral are radii or either circle A or circle B, which equal to a length of b/2. Thus, ABCD is a square of sides b/2. Diagonal AB is of the length sqrt(2)*b/2 = b/sqrt(2).

Also, len(AB) = a = b/sqrt(2). Thus, b/a = sqrt(2). Ta-da!
Insufficient data to prove ABCD is a square. Agreed ACB and BDA are right angles, but CAD and DBC (well, this is my fav - Death By Chocolate!) being right angles is an unproved assumption.
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Prasad Ajinkya

@prasad-aSUfhP • Apr 26, 2007

NiLaY, its not an assumption, ABCD has to be a square. This is the logic -

Since AC = AD = b/2 (radii of the circle) and BC = BD = b/2 (radii of the other circle)
Thus, it is established that ABCD is a paralellogram (a rhombus infact)

Now, its common sense that if two opposite angles of a parallelogram are right angles, the other two are also right angles (do the math 😀 ). Thus, our rhombus is a square.

QED.
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NiLaY

@nilay-lpoAmu • Apr 26, 2007

Please refer to the image below (common sense). Though, the answer might be correct, I am afraid solution is not. 😕
We need to consider the footballs in this puzzle.

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crook

@crook-0PFkJv • May 1, 2007

Where did you get that image NiLaY 😉 The puzzle is turning out to be more difficult that I thought 😁
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NiLaY

@nilay-lpoAmu • May 1, 2007

I did not GET 😒 it, I combined a few autoshape circles/rectangles/lines in M$ Word to make this.
By the way, anyone up with another solution?
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Kaustubh Katdare

@thebigk • Jun 7, 2007

Here we go -

[FONT=Verdana, Arial, Helvetica, sans-serif]We can therefore solve this puzzle with the help of the theorem of Pytagora:
AA'2 = AB2 + BA'2
a2 = (b/2)2+ (b/2)2 = 2(b/2)2
Simplifying: a = b2/2[/FONT]
[FONT=Verdana, Arial, Helvetica, sans-serif]So: b/a = b/(b2/2) = 2[/FONT]

[FONT=Verdana, Arial, Helvetica, sans-serif](as posted on #-Link-Snipped-# )[/FONT]

[FONT=Verdana, Arial, Helvetica, sans-serif]Good job, Kidakaka !
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[FONT=Verdana, Arial, Helvetica, sans-serif]-The Big K-
[/FONT]

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