CIRCLES https://gmatclub.com/forum/download/file.php?id=10392 This post is a part of GMAT MATH BOOK ] created by: Bunuel edited by: bb , walker https://gmatclub.com/forum/download/file2.php?id=23736 -------------------------------------------------------- Get The Official GMAT Club's App - GMAT TOOLKIT 2 . The only app you need to get 700+ score! iOS App ] Android App ] -------------------------------------------------------- Definition A line forming a closed loop, every point on which is a fixed distance from a center point. Circle could also be defined as the set of all points equidistant from the center. https://gmatclub.com/forum/download/file.php?id=10343 Center -a point inside the circle. All points on the circle are equidistant (same distance) from the center point. Radius - the distance from the center to any point on the circle. It is half the diameter. Diameter -t he distance across the circle. The length of any chord passing through the center. It is twice the radius. Circumference - the distance around the circle. Area - strictly speaking a circle is a line, and so has no area. What is usually meant is the area of the region enclosed by the circle. Chord - line segment linking any two points on a circle. Tangent -a line passing a circle and touching it at just one point. The tangent line is always at the 90 degree angle (perpendicular) to the radius of a circle. Secant A line that intersects a circle at two points. \pi In any circle, if you divide the circumference (distance around the circle) by it's diameter (distance across the circle), you always get the same number. This number is called Pi and is approximately 3.142. https://gmatclub.com/forum/download/file.php?id=10344 • A circle is the shape with the largest area for a given length of perimeter (has the highest area to length ratio when compared to other geometric figures such as triangles or rectangles) • All circles are similar • To form a unique circle, it needs to have 3 points which are not on the same line. Circumference, Perimeter of a circle Given a radius r of a circle, the circumference can be calculated using the formula: Circumference=2\pi{r} If you know the diameter D of a circle, the circumference can be found using the formula: Circumference=\pi{D} If you know the area A of a circle, the circumference can be found using the formula: Circumference=\sqrt{4\pi{A}} Area enclosed by a circle Given the radius r of a circle, the area can be calculated using the formula: Area={\pi}r^2 If you know the diameter D of a circle, the area can be found using the formula: Area=\frac{{\pi}D^2}{4} If you know the circumference C of a circle, the area can be found using the formula: Area=\frac{C^2}{4{\pi}} Semicircle Half a circle. A closed shape consisting of half a circle and a diameter of that circle. https://gmatclub.com/forum/download/file.php?id=10356 • The area of a semicircle is half the area of the circle from which it is made: Area=\frac{{\pi}r^2}{2} • The perimeter of a semicircle is not half the perimeter of a circle. From the figure above, you can see that the perimeter is the curved part, which is half the circle, plus the diameter line across the bottom. So, the formula for the perimeter of a semicircle is: Perimeter=\pi{r}+2r=r(\pi+2) • The angle inscribed in a semicircle is always 90°. • Any diameter of a circle subtends a right angle to any point on the circle. No matter where the point is, the triangle formed with diameter is always a right triangle. https://gmatclub.com/forum/download/file.php?id=10346 Chord A line that links two points on a circle or curve. https://gmatclub.com/forum/download/file.php?id=10347 • A diameter is a chord that contains the center of the circle. • Below is a formula for the length of a chord if you know the radius and the perpendicular distance from the chord to the circle center. This is a simple application of Pythagoras' Theorem. Length=2\sqrt{r^2-d^2} , where r is the radius of the circle, d is the perpendicular distance from the chord to the circle center. • In a circle, a radius perpendicular to a chord bisects the chord. Converse: In a circle, a radius that bisects a chord is perpendicular to the chord, or In a circle, the perpendicular bisector of a chord passes through the center of the circle. Angles in a circle An inscribed angle is an angle ABC formed by points A, B, and C on the circle's circumference. https://gmatclub.com/forum/download/file.php?id=10348 • Given two points A and C, lines from them to a third point B form the inscribed angle ∠ABC. Notice that the inscribed angle is constant. It only depends on the position of A and C. • If you know the length L of the minor arc and radius, the inscribed angle is: Angle=\frac{90L}{\pi{r}} A central angle is an angle AOC with endpoints A and C located on a circle's circumference and vertex O located at the circle's center. A central angle in a circle determines an arc AC. https://gmatclub.com/forum/download/file.php?id=10349 • The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. https://gmatclub.com/forum/download/file.php?id=10350 • An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees). Arcs and Sectors A portion of the circumference of a circle. https://gmatclub.com/forum/download/file.php?id=10351 • Major and Minor Arcs Given two points on a circle, the minor arc is the shortest arc linking them. The major arc is the longest. On the GMAT, we usually assume the minor (shortest) arc. • Arc Length The formula the arc measure is: L=2\pi{r}\frac{C}{360} , where C is the central angle of the arc in degrees. Recall that 2\pi{r} is the circumference of the whole circle, so the formula simply reduces this by the ratio of the arc angle to a full angle (360). By transposing the above formula, you solve for the radius, central angle, or arc length if you know any two of them. • Sector is the area enclosed by two radii of a circle and their intercepted arc. A pie-shaped part of a circle. • Area of a sector is given by the formula: Area=\pi{r^2}\frac{C}{360} , where: C is the central angle in degrees. What this formula is doing is taking the area of the whole circle, and then taking a fraction of that depending on the central angle of the sector. So for example, if the central angle was 90°, then the sector would have an area equal to one quarter of the whole circle. Power of a Point Theorem Given circle O, point P not on the circle , and a line through P intersecting the circle in two points. The product of the length from P to the first point of intersection and the length from P to the second point of intersection is constant for any choice of a line through P that intersects the circle. This constant is called the "power of point P" . If P is outside the circle: https://gmatclub.com/forum/download/file.php?id=10352 PA*PD=PC*PB=Constant - This becomes the theorem we know as the theorem of intersecting secants. If P is inside the circle: https://gmatclub.com/forum/download/file.php?id=10657 PA*PD=PC*PB=Constant - This becomes the theorem we know as the theorem of intersecting chords. Tangent-Secant https://gmatclub.com/forum/download/file.php?id=10354 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. Two tangents https://gmatclub.com/forum/download/file.php?id=10355 Should both of the lines be tangents to the circle, point A coincides with point D, point C coincides with point B, and the theorem still applies: PA*PD=PC*PB=Constant PA^2=PC^2 PA=PC Official GMAC Books: The Official Guide, 12th Edition: DT #36; PS #33; PS #160; PS #197; PS #212; DS #42; DS #96; DS #114; DS #117; DS #160; DS #173; The Official Guide, Quantitative 2th Edition: PS #33; PS #141; PS #145; PS #153; PS #162; DS #22; DS #58; DS #59; DS #95; DS #99; The Official Guide, 11th Edition: DT #36; PS #30; PS #42; PS #100; PS #160; PS #206; PS #229; DS #23; DS #76; DS #86; DS #136; DS #152; Generated from GMAT ToolKit ] https://gmatclub.com/forum/download/file2.php?id=23736 -------------------------------------------------------- Get The Official GMAT Club's App - GMAT TOOLKIT 2 . The only app you need to get 700+ score! iOS App ] Android App ] -------------------------------------------------------- Math_cir_3.png Math_cir_13.png Math_cir_12.png Math_cir_10.png Math_cir_9.png Math_cir_8.png Math_cir_7.png Math_cir_6.png Math_cir_5.png Math_cir_4.png Math_cir_2.png Math_cir_1.png Math_icon_circles.png Math_cir_11.png

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Math: Circles

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galiya

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26 Jun 2011, 11:55

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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?

fivedaysleft

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Galiya

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.

Asher

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23 Jul 2011, 21:40

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Thanks Bunuel.. love your explanations... very simple, clear and easy to comprehend.. Kudos

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This is probably my worst area in all of gmat. This helps! thanks!

navi19

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20 Nov 2011, 06:43

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Wow.... I've just started and U've made me fall in love with Circles all over... Thank you...

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16 Aug 2012, 11:42

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Bunuel

chauhan2011

• 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?

goutamread

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28 Oct 2012, 11:52

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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 :

\(A=\frac{1}{2}r^2*(\frac{pi*center angle}{180}-sin(\frac{pi*center angle}{180}))\)

Bunuel

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11 Jul 2013, 00:06

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Bumping for review*.

*New project from GMAT Club!!! Check HERE

All Theory Topics: search.php?search_id=tag&tag_id=351
MATH BOOK: gmat-math-book-87417.html

naiduashwini

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20 Jul 2013, 23:16

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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?

Bunuel

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21 Jul 2013, 01:55

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naiduashwini

____________
Yes, it does.

madn800

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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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11 Dec 2013, 21:00

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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°."

Bunuel

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12 Dec 2013, 02:52

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catalysis

Check the diagram below:

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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12 May 2014, 12:05

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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.

Bunuel

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13 May 2014, 00:57

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gaurav1418z

Some aspects of this properties definitely could be helpful when solving GMAT questions.

saraafifiahmed

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Hi Brnuel ,

Would you please help me with this "easy " questions.

Triangle ABC is inscribed in a circle, such that AC is a diameter
of the circle (see figure). If AB has a length of 8 and BChas a
length of 15, what is the circumference of the circle?

Aren't we supposed to - after getting AC= 17 ( which is the diameter) to calculate the circumference as 2 π R .

Is not the r supposed to be 17/2 ?. Thanks.

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17 May 2015, 02:38

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Excellent work. All the important things in just one post.
Power of point theorem is vague to me. How do we know that the product of two point of intersections is constant? Could someone explain please?

Thanks

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04 Aug 2015, 09:58

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Bunnel - on posts for triangle and circle i found the following- which one is correct ?

For a given perimeter equilateral triangle has the largest area.
A circle is the shape with the largest area for a given length of perimeter

Thanks
Prasad

Bunuel

gaurav1418z

Some aspects of this properties definitely could be helpful when solving GMAT questions.

Bunuel

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16 Aug 2015, 10:03

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prasadcp

Bunuel

gaurav1418z

Some aspects of this properties definitely could be helpful when solving GMAT questions.

Both.

The first property is for triangles only. Meaning that for a given perimeter of a triangle, equilateral triangle has the largest area.

nishantdoshi

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24 Jan 2016, 00:35

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the Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle.

Image

• An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

in the image there's an angle alpha which is outside the circle which could be found out easily if we are given its supplementary angle,
hey bunuel what i cant understand is the relation between the central angle and the inscribed angle
can you please help me out?

i cant get the image pasted here :/