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Circle A, center X. XB is the radius. There is a chord which

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Circle A, center X. XB is the radius. There is a chord which [#permalink] New post 21 Mar 2008, 23:36
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Circle A, center X. XB is the radius. There is a chord which intersects XB. D is the point of intersection between XB and AC. BD = 2, AC = 12, XDA = 90 degrees. What is the cirle's area?
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Re: PS (geometry) [#permalink] New post 22 Mar 2008, 00:16
az780 wrote:
Circle A, center X. XB is the radius. There is a chord which intersects XB. D is the point of intersection between XB and AC. BD = 2, AC = 12, XDA = 90 degrees. What is the cirle's area?


My computer does not support drawing, pls get it by yoursefl
1. XDA = 90 degr. so XB perpendicular with AC, and so D is mid point of AC, that is AD=DC =6

2 Triangle XDA: (XA)^2 = (XD +2)^2 = (XD)^2 + (DA)^2 = (XD)^2 + 36
==> XD=8 --> r = 8+2 = 10 --> Area of cirle = 100pi

EXPECT A LESS TIME-CONSUMING APPROACH!
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Re: PS (geometry) [#permalink] New post 22 Mar 2008, 00:34
Expert's post
Another way:

1. ABY is geometrically similar to DBA

2. \frac{AB}{2}=\frac{BY}{AB} -->BY=\frac{AB^2}{2}=\frac{6^2+2^2}{2}=20

3. Area = \frac{\pi*BY^2}{4}=100\pi
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Re: PS (geometry) [#permalink] New post 13 May 2008, 12:09
I think I need some serious work in geometry.

Could someone please explain the following...
2 Triangle XDA: (XA)^2 = (XD +2)^2 = (XD)^2 + (DA)^2 = (XD)^2 + 36
==> XD=8 --> r = 8+2 = 10 -->

or walker's method? Thanks
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Re: PS (geometry) [#permalink] New post 13 May 2008, 16:25
walker wrote:
Another way:

1. ABY is geometrically similar to DBA

2. \frac{AB}{2}=\frac{BY}{AB} -->BY=\frac{AB^2}{2}=\frac{6^2+2^2}{2}=20

3. Area = \frac{\pi*BY^2}{4}=100\pi


the first answer is straight forward Pythagorean theorem.
i like the walkers answer, but don’t really get it.

walker could you please shed some light!
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Re: PS (geometry) [#permalink] New post 13 May 2008, 18:40
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the first step is the most important in my approach:

the triangles DBA and ABY are right triangles, because angles ADB and YAB are right angles.
the triangles DBA and ABY have the same angle ABY=ABD and, therefore, third angles in triangles are equal: AYB=DAB
So,
1. ABY is geometrically similar to DBA

Next step is easer, because I only use the formula for similar triangles:

\frac{BY}{AB}=\frac{AY}{AD}=\frac{AB}{BD}
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Re: PS (geometry) [#permalink] New post 14 May 2008, 08:47
Thanks Walker.

If anyone is wondering how we know that angle BAY is right, look up Thales' theorem. Until I looked at walker's method, I had no idea this theorem existed. I really don't remember this from high school. Did I simply forget it, or is it not taught in US public schools?

Also, if anyone is confused by the first (simple) method, the trick is to recognize that XB = XA = XD + DB
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Re: PS (geometry) [#permalink] New post 14 May 2008, 19:34
sondenso wrote:
az780 wrote:
Circle A, center X. XB is the radius. There is a chord which intersects XB. D is the point of intersection between XB and AC. BD = 2, AC = 12, XDA = 90 degrees. What is the cirle's area?


My computer does not support drawing, pls get it by yoursefl
1. XDA = 90 degr. so XB perpendicular with AC, and so D is mid point of AC, that is AD=DC =6

2 Triangle XDA: (XA)^2 = (XD +2)^2 = (XD)^2 + (DA)^2 = (XD)^2 + 36
==> XD=8 --> r = 8+2 = 10 --> Area of cirle = 100pi

EXPECT A LESS TIME-CONSUMING APPROACH!


sorry. i dont see how you get XD=8.
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Re: PS (geometry) [#permalink] New post 15 May 2008, 06:41
seongbae wrote:
sondenso wrote:
az780 wrote:
Circle A, center X. XB is the radius. There is a chord which intersects XB. D is the point of intersection between XB and AC. BD = 2, AC = 12, XDA = 90 degrees. What is the cirle's area?


My computer does not support drawing, pls get it by yoursefl
1. XDA = 90 degr. so XB perpendicular with AC, and so D is mid point of AC, that is AD=DC =6

2 Triangle XDA: (XA)^2 = (XD +2)^2 = (XD)^2 + (DA)^2 = (XD)^2 + 36
==> XD=8 --> r = 8+2 = 10 --> Area of cirle = 100pi

EXPECT A LESS TIME-CONSUMING APPROACH!


sorry. i dont see how you get XD=8.


(XA)^2 = (XD +2)^2 = (XD)^2 + (DA)^2 = (XD)^2 + 36

take one thing at a time.

(XA)^2 = (XD +2)^2 (Because XA and (XD + 2) are both equal to the radius of the circle)


(XD)^2 + (DA)^2 (Imagine a triangle ADX, with angle ADX a right angle. This is the pythagorean theorem. Leg AD squared plus leg XD squared equals hypotenuse AX squared. This equals XD squared plus 36.

So AX^2 equals both (XD +2)^2 and (XD)^2 + 36, and since (XD +2)^2 = (XD)^2 + 36, we can solve for XD

(XD)^2 + 2 XD + 2 XD + 4 = (XD)^2 + 36
4 XD = 32
XD = 8
Re: PS (geometry)   [#permalink] 15 May 2008, 06:41
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