earth@work wrote:A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.
Cud anyone help me with this, thanks?
While this question is worded ambiguously (and therefore can't be a real GMAT question), I'm pretty sure we want to find the straight line distance between 3 points situatied equidistant from each other around a circle.
In other words, if you draw a circle and plot 3 points around the circumference which are 120 degrees apart from each other, what's the length of a side of the triangle formed by those 3 points.
I'm sure there's a fancy formula to calculate the side of an equilateral triangle inscribed in a circle, but it's certainly not a formula you need for test day, so let's work this out using special triangles.
Let's focus on two of the boys and the centre of the circle. The degree arc at the centre is 120 degrees (360/3) and, since two of the sides of the triangle are radii of the circle, it's isosceles. So, we have a 120/30/30 triangle.
Now let's cut that triangle in half. The angle at the centre of the circle is 60 degrees and the angle near the boy is 30 degrees, which leaves us with 90 degrees for the third angle. In other words, we have a 30/60/90 triangle with the radius of the circle as the hypotenuse.
The ratio of the sides of a 30/60/90 is x/xroot3/2x. In this question, the radius is 20, so:
2x = 20
x = 10
xroot3 = 10root3
The side we care about is the one opposite the 60 degree angle, the xroot3 side.
We originally cut our 120/30/30 triangle in half. In order to get the full length of the side, we need to double our previous result.
2 * 10root3 = 20root3... there's our answer.
(As an aside, if you want your fancy formula for the length of a side of an equilateral triangle inscribed inside a circle, it's radius*root3 (as we just proved).)
Of course, this would have been much easier with a diagram, which is why one should always draw out geometry problems on one's scrap paper!
