In the diagram below, P and Q are the centers of the two circles. What

Question

In the diagram below, P and Q are the centers of the two circles. What is the area of triangle XYZ?

(1) Radius of each circle is 6.
(2) XY = 2.

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2017-07-18_1034.png [ 11.9 KiB | Viewed 5813 times ]

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Official Answer

B

sananoor · Answer

Is it C? from statement 1 we can see that the triangle is equilateral as radius of two circles are same and xz and yz will have same value, but i am not sure about xy statement 2 gives us the value of one side so if triangle is equilateral we can find the area of triangle.

rohit8865 · Answer

Well .....

i will mark E as answer .....

i cannot find any way to find values of yz and xz....

mohshu · Answer

stat2: lets take this statement first,,,just the length of one segment given,,,not suff..

stat1 : inspite of drawing/extending the segments ont he figure,,this info did not prove useful to me..

ans E...

jedit · Answer

It should be  C .

Statement 1: Insufficient

Knowing the radius tells us nothing about the points of intersection.

Statement 2: Insufficient

It tells us nothing about the position of points  Z  and  X  relative to the centers of the circles and points  Y  and  X .

Statement 1 + 2: Sufficient  - Bear with me as it may not be easy to understand.

Circle P:

- We know that arc  ZQV  is  120  degrees since circle  P  is being carved by a circle  Q  of equal radius.
- Angle that point  Y  creates with points  Z  and  V  is half of the central angle  ZPV   (central angle theorem) .
- This makes angle  Y = 120*\frac{1}{2} = 60  degrees.

See figure 1.

Circle Q:

- We know that point  Q  splits angle  Y  exactly in half (project line  YZ  further into the circle Q where it hits the opposite end internally to understand. You can shift Y as well on the circumference of circle P).
- Thus radius  QX  meets the chord  YV  exactly where  XY = YZ  ( Z  is already on radius of  Q ).

See figure 2 to understand

- We now have two sides  XY  and  XZ  equal and their central angle  60  degrees.
- This is all we need since this makes it an equilateral triangle (draw it to understand better). We can now apply the formula for area of equilateral triangle  XY^2 * \sqrt{3}/4 . - This results in  \sqrt{3}

mohshu · Answer

jedit  im not able to understand the above statement...

jedit · Answer

when two circles of equal radius overlap and intersect each other's centers, the arc of overlapping area is 120 degrees. this can be proven easily by connecting centers and center to points of intersection. This makes two equilateral triangles and central angle is sum of individual angles of two triangles. I am on my mobile or I would draw the picture for you.

Sent from my Redmi Note 4 using  GMAT Club Forum mobile app

mohshu · Answer

jedit  im not able to understand the above statement... 
when two circles of equal radius overlap and intersect each other's centers, the arc of overlapping area is 120 degrees. this can be proven easily by connecting centers and center to points of intersection. This makes two equilateral triangles and central angle is sum of individual angles of two triangles. I am on my mobile or I would draw the picture for you.

Sent from my Redmi Note 4 using  GMAT Club Forum mobile app

nope diagram not required,,, got ur point,,,thanks

gps5441 · Answer

Which property of circle and ∆ have used here to prove that the ∆ xyz is equilateral. I have understood how angle xyz is 60°.

Posted from my mobile device

Shobhit7 · Answer

Ans B:
In Triangle PZQ- all sides are same and equal to the radius. Hence, Angle ZPQ= 60 degree.
Similarly, In Triangle PQV- Angle QPV= 60 degree.
So, Angle ZPV = Angle ZQV = 60+60 = 120 degree.
Therefore, Angle ZYV (or ZYX) = Angle Half of Angle ZPV (Inscribed Angle is half of Central Angle) = 60 degrees.
Now, Angle ZXV = 120 degrees. (Half of Major Angle ZQV= 240 degree).
So, Angle YXZ = 180-120= 60 degree.
That makes Triangle XYZ an Equilateral Triangle.
St. 2 gives us length of side XY =2.
Therefore, Area of Triangle XYZ= (√3/4) x (XY) square = √3

pikolo2510 · Answer

Hi Shobhit7 Angle ZXV = 120 degrees. (Half of Major Angle ZQV= 240 degree) Can you explain this part again? I didn't get this! Also it would be great if you can post a picture of the diagram with the calculations you did. Thanks :-)

Shobhit7 · Answer

In circle with centre Q, points Z and V create two arcs, minor arc of 120 degree and major arc of angle 240 degree.

Angle ZXV is inscribed angle to major arc and hence measures 120 degree (Inscribed angle is half of Central Angle)

aserghe1 · Answer

picture.png I think we can solve this question without doing any calculations. (1) Both circles have same radius. Looking at the attached picture (ignore the part about XY; that's for Statement 2), we can vary the triangle, and thus can vary the area of the triangle. Not sufficient. (2) We're given that XY=2. Even if we don't know how to find the area of triangle XYZ, we know that there can only be 1 triangle XYZ where XY=2 (see picture). Thus, there is one area, which can be found. Sufficient. Answer: B

bumpbot · Answer

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In the diagram below, P and Q are the centers of the two circles. What

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