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Geometry - Circles and Chords

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shankar.ashwin: Legendary Member; Posts: 966; Joined: Sat Jan 02, 2010 8:06 amFollowed by: 21 members; Thanked: 230 times

by shankar.ashwin » Sun Nov 06, 2011 4:44 am

In the figure, a, b, c, d, e, f, g, h, i, and j are chords of the circle. Which two chords are parallel to each ï¿¼other?

(A) a and f
(B) b and g
(C) c and h
(D) d and i
(E) e and j

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Source: Beat The GMAT — Problem Solving | Sun Nov 06, 2011 4:44 am

GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members; Location: Austin, TX; Thanked: 236 times; GMAT Score:770

by GmatMathPro » Sun Nov 06, 2011 9:19 am

Ans: E. e and j

Imagine you're walking around this decagon counter-clockwise. As you walk down chord j, you have to make a 35 degree turn to get to chord i. Then you have to make a 32 degree turn to get to chord h. But now, you're facing in the same direction as if you had just made a 32+35=67 degree turn from chord j. As you continue to walk from chord to chord, the degrees of your turns continue to add up, so going all the way from j to e, you have to turn 35+32+45+36+32=180 degrees. So, you are on a different spot on the decagon, but you've essentially turned 180 degrees when you walk on chord e compared to when you were walking on chord j, so the chords are parallel. Note that this kind of reasoning can be used to show that the exterior angles of a polygon add up to 360: if you go all the way around the polygon, you're back facing the way you were at the beginning, so you've made one full 360 degree turn.

Pete Ackley
GMAT Math Pro
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