Math: Circles

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

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

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?

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 :

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

Bumping for review*.

*New project from GMAT Club!!! Check HERE

Tangent-Secant


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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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Yes, it does.

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.

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:

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.

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.

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.

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

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

Both.

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

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


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• 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 :/