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Math: Circles

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Circles inscribed triangles [#permalink] New post 26 Jun 2011, 10:55
Hi all!
Help me please with one task concerning inscribed triangles and circles :roll:
here it is:
Circle A,centre X. XB is the radius. There is a chord AC which intersects XB. D is the point of intersection between XB and AC. BD=2;AC=12;XDA= 90 degrees. What is the circles area?
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Re: Circles inscribed triangles [#permalink] New post 01 Jul 2011, 12:07
Galiya wrote:
Hi all!
Help me please with one task concerning inscribed triangles and circles :roll:
here it is:
Circle A,centre X. XB is the radius. There is a chord AC which intersects XB. D is the point of intersection between XB and AC. BD=2;AC=12;XDA= 90 degrees. What is the circles area?



By basic property of the circle,
the radius bisects the chord AC . ie CD equals AC/2 ie 6

now see, radius XB=XD+BD ie r=XD+2 ie XD=r-2

now concerning Triangle XCD, angle XDC= XDA =90 so pythagorean theorem is applicable
so, sq(XC)=sq(DC)+sq(XD)

plug in values you get sq(r)=sq(r-2)+sq(6)
which gives r=10

now area=pi*r*r=100*pi

Hope it helps.
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Re: Math: Circles [#permalink] New post 23 Jul 2011, 20:40
Thanks Bunuel.. love your explanations... very simple, clear and easy to comprehend.. Kudos
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Re: Math: Circles [#permalink] New post 13 Nov 2011, 22:30
This is probably my worst area in all of gmat. This helps! thanks!
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Re: Math: Circles [#permalink] New post 20 Nov 2011, 05:43
Wow.... I've just started and U've made me fall in love with Circles all over... Thank you...
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Re: Math: Circles [#permalink] New post 16 Aug 2012, 10:42
Bunuel wrote:
chauhan2011 wrote:
• If you know the length of the minor arc and radius, the inscribed angle is: 90L/nr

Please correct me if i am wrong but i think the formula should be : 180L/nr


If you know the length L of the minor arc and radius, the inscribed angle is: Inscribed \ Angle=\frac{90L}{\pi{r}}.

The way to derive the above formula:

LengAngle=\frac{180L}{\pi{r}}[/m] --> Inscribed \ Angle=\frac{90L}{\pi{r}}.

Hope it helps.



Hello, quick conceptual question

The circle represented by the equation x^ 2 + y^ 2 =1 is centered at the origin and has the radius of r= √1 = 1

What is the correlation between the function and the radius for a circle?
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Re: Math: Circles [#permalink] New post 28 Oct 2012, 10:52
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A secant\chord (except diameter) to a circle divides circle into two region - minor and major. The area of minor region can be calculate by determining area of minor sector minus triangle. Also the direct formula to calculate minor region area is :

A=\frac{1}{2}r^2*(\frac{pi*center angle}{180}-sin(\frac{pi*center angle}{180}))
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Re: Math: Circles [#permalink] New post 10 Jul 2013, 23:06
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Re: Math: Circles [#permalink] New post 20 Jul 2013, 22:16
Tangent-Secant


Image

Should one of the lines be tangent to the circle, point A will coincide with point D, and the theorem still applies:

PA*PD=PC*PB=Constant

PA^2=PC*PB=Constant - This becomes the theorem we know as the theorem of secant-tangent theorem.


Does this hold true if CB was the diameter of the circle?
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Re: Math: Circles [#permalink] New post 21 Jul 2013, 00:55
Expert's post
naiduashwini wrote:
Tangent-Secant


Image

Should one of the lines be tangent to the circle, point A will coincide with point D, and the theorem still applies:

PA*PD=PC*PB=Constant

PA^2=PC*PB=Constant - This becomes the theorem we know as the theorem of secant-tangent theorem.


Does this hold true if CB was the diameter of the circle?

____________
Yes, it does.
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Re: Math: Circles [#permalink] New post 01 Dec 2013, 04:04
Can somebody explain the properties of a cyclic quadrilateral. Also, do the same properties hold good for a cyclic quadrilateral inscribed in a semicircle with one of its sides being the diameter of that semicircle.
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Re: Math: Circles [#permalink] New post 11 Dec 2013, 20:00
Thanks so much!!

Quick question - under the "Semicircles" section, could you clarify or show a picture of what this means? I don't understand how it would always be true.

"• The angle inscribed in a semicircle is always 90°."
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Re: Math: Circles [#permalink] New post 12 Dec 2013, 01:52
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catalysis wrote:
Thanks so much!!

Quick question - under the "Semicircles" section, could you clarify or show a picture of what this means? I don't understand how it would always be true.

"• The angle inscribed in a semicircle is always 90°."


Check the diagram below:
Image
Angle ABC is inscribed in semicircle, thus angle B is 90 degrees.

It's the same as the following property: a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle.

Hope it's clear.
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Re: Math: Circles [#permalink] New post 12 May 2014, 11:05
Bunuel, first of all thanks for another amazing post.

Wanted to check, there are a few concepts in this thread like secant, chord, point theorem etc. Are they tested in GMAT? And i have same question with your other quant concept threads.
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Re: Math: Circles [#permalink] New post 12 May 2014, 23:57
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gaurav1418z wrote:
Bunuel, first of all thanks for another amazing post.

Wanted to check, there are a few concepts in this thread like secant, chord, point theorem etc. Are they tested in GMAT? And i have same question with your other quant concept threads.


Some aspects of this properties definitely could be helpful when solving GMAT questions.
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PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Math: Circles   [#permalink] 12 May 2014, 23:57
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