Ps Geometry

Question

A circle with center o is inscribed in the square wxyz. Point P is where the circle and line segment ZX intersect. If PX = 1, which of the following is the closest to the circumference?

A 2 pi
B 3 pi
C 4 pi
D 5 pi
E 6 pi

trivikram · Answer

6PI

let x be the side of the square

sqrt(2) = 2 + 2r (2 'cos 2 times PX + daimeter)

x =2/(sqrt(2) -1) 

Calculate for radius and get 6PI as the circumeference

hobbit · Answer

agree with the formula and approach, but result is actually 5pi and not 6.

ncp · Answer

Agree with Hobbit. I get r=1/(1.414-1).

Circumference=2PIr=approx 5PI

anindyat · Answer

Yes ..same approach but result = 2 * pi * (1 + sqrt2) = 5*pi

gk3.14 · Answer

2 cos - using trig?
Is it possible without it?

anindyat · Answer

Say r is the radius
Diameter = 2r
Each side of the square = 2r
Diagonal = sqrt   = sqrt (8r^2)
PX = 1, OX = r + 1, Diagonal = 2 (r +1)

 sqrt (8r^2) = 2(r + 1)
=> 2sqrt2 * r - 2r = 2
=> r(sqrt2 -1) = 1
=> r = 1/( sqrt2 -1) = ( sqrt2 +1)

Circumference = 2 pi r = 2 pi (sqrt2 +1) = 2 * pi * (1.4 +1) = 4.8 * pi ~ 5pi

MBAlad · Answer

What is wrong with this method??

Circle diameter = D
Side of square = D
Diagonal ZX = D+2
Diagonal ZX = Dsqrt(2) from ratio of sides or from D^2+D^2=2D^2

So Dsqrt(2) = D+2 Square both sides:
2D^2 = D^2+4
D^2 = 4
D=2
???????? What went wrong?????????

anindyat · Answer

You are working late these days ........

MBAlad · Answer

Brilliant - dont you just hate the way you stare at a problem for ages and you just cant see what is wrong!

Thanks anindyat!

mich120 · Answer

Can someone explain further? I get to the diagonal being 2r+2, but am unsure where to go from there. For some reason I am suck on this problem!

Thanks in advance!

ocprep · Answer

Can someone explain it to me like I'm a 10 yr old? Need step by step. I get how you get the diagonal, the sides then I'm lost. I tried the pathagoreum form but doesn't seem to lead to 5pi. can you use the pathagoreum to solve this?

mich120 · Answer

1) Length of diagonal would be 2r + 2 (where r is the radius of the circle and 2 is PX+length of Z to the end of the circle)
2) The rules of a 45, 45, 90 triangle indicate that the length across from the 90 degree angle (which in this case is the diagonal of the circle) must be length x*sqrt(2) in relation to the sides of the square, each at x. We know that the sides of the square are 2r long. 
3) We can conclude that 2r*sqrt(2) = 2r + 2
4) Then solve for r:
 2r*sqrt(2) = 2(r + 1) 
Then divide both sides by 2:
 r*sqrt(2) = (r + 1)
Then get the r's to one side:
 r*sqrt(2)-r = 1
Pull out the r: 
 r(sqrt(2)-1) = 1
Divide to get r alone:
 r=1/(sqrt(2)-1) = 1/(1.414-1)
so... r is approx. 2.5
making the circumference of the circle approx. 5Pi

Hope this helps!

ocprep · Answer

Kudos mich...it was driving me crazy all day.

pmenon · Answer

so we have the following:

d*root(2) = 1+d+1, where d is the diameter of the circle, and circumference of a circle is pi*diameter

for a square, the diagonal of the square is side*root(2). In this case, we have side*root(2) = d*root(2)=2+d

d = 2/(root(2)-1), and root(2) is approx 1.4, so we have d = 2/0.4 = 5(approx)

Circumference is pi*d = pi*5

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