It asks for the length which is two pi.
you're thinking of the measure of the chord, which is 40.
I am taking a practice test from the company that created the GMAT test. One geometry question has me baffled, because I get 40 and the correct answer according to the test is 2Pi (option 1). I have uploaded a picture of the question because I don't know how if it's possible to draw the circle and the lines that are in the circle on this forum. Can you please take a look and let me know what you think? Thank you.
Angle DBA is a right angle as it is the angle in a semi circle.
Angle DAB is 55 degrees as it is the thrid angle in a triangle who's other
angles are 90 and 35 degrees.
Angle CDA= Angle DAB = 55 degrees(I leave the explanation of this to the reader)
So Angle CDB=20 degrees.
Now Angle BOC is twice Angle CDB, as the first is the angle subtended at the centre
by the minor arc BC, and the second is the angle subtended by the same arc from a point
on the major arc BC.
40 degrees is , so the arc length BC is the radius of the
circle times the angle .
To get the length of minor arc PQ we need to know the central angle subtending that PQ.
Since OR and PQ are paralle, then angle QPR equals angle ORP equals 35 deg.
Draw radii to P and Q. Call the center of the circle, point C.
Triangle PCR is isosceles, because its two sides, PC and RC, are a radius each.
So, angle RPC = angle PRC = 35 deg.
In triangle PCQ, angle QPC = 35 +35 = 70 deg.
Triangle PCQ is isosceles also because PC and QC are both radii.
So, angle PQC = angle QPC = 70 deg.
Hence, the 3rd angle, angle PCQ = 180 -(70 +70) = 40 deg.
Angle PCQ is also the central angle of minor arc PQ.
arc = (radius)(central angle in radians)
minor arc PQ = (18/2)(40 *pi/180)
minor arc PQ = 360pi /180
minor arc PC = 2pi units long
Hence, letter a) 2pi, is the answer.
arc length = (radius)(central angle in radians).
Then use proportion.
The central angle for the minor arc PQ is 40deg.
The "central angle" for the whole circle is 360 deg.
(minor arc PQ)/(40deg) = (18pi)/(360deg)
Multiply both sides by 40deg,
minor arc PQ = (40deg)(18pi) /(360deg) = (18pi)/9 = 2pi
One thing in Math, there are many ways to arrive at the same answer.
Isn't Math fun?
Area of sector = (1/2)(radius)(arc) ---------"similar" to that of a triangle.
Area of sector = (1/2)(radius)[(radius)(central angle in radians)]
Area of sector = (1/2)(r^2)(central angle in radians)
Or, use proportion also.
(area of sector)/(central angle) = (pi*r^2)/(360deg)