problem10

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Captchar: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sun Dec 16, 2012 4:04 am

by Captchar » Sun Dec 23, 2012 11:02 am

Dan and Karen who live 10 miles apart meet at a cafe that is directly north of Dan's house and directly east of Karen's house. If the cafe is 2 miles closer to Dan's house than to Karen's house, how many miles is the cafe from Karen's house?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

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Source: Beat The GMAT — Problem Solving | Sun Dec 23, 2012 11:02 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sun Dec 23, 2012 11:14 am

Captchar wrote:Dan and Karen who live 10 miles apart meet at a cafe that is directly north of Dan's house and directly east of Karen's house. If the cafe is 2 miles closer to Dan's house than to Karen's house, how many miles is the cafe from Karen's house?

Refer to the figure above.
C is the Cafe, D is Dan's house and K is Karen's house.

Hence, the hypotenuse, KD = 10
And if KC = x, CD = (x - 2)

So, xÂ² + (x - 2)Â² = 10Â² = 100
--> xÂ² + xÂ² - 4x + 4 = 100
--> 2xÂ² - 4x - 96 = 0
--> xÂ² - 2x - 48 = 0
--> xÂ² - 8x + 6x - 48 = 0
--> (x + 6)(x - 8) = 0

Hence, either x = -6 or x = 8
As x cannot be negative, x = 8

The correct answer is C.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sun Dec 23, 2012 11:18 am

Anurag@Gurome wrote:So, xÂ² + (x - 2)Â² = 10Â²

After this point the problem can be solved easily without actually solving the quadratic equation, if we remember that (6, 8, 10) constitutes a Pythagorean triplet.

Hence, 6Â² + 8Â² = 10Â² and (8 - 6) = 2

Hence, the cafe is 8 miles from Karen's house.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Dec 23, 2012 3:52 pm

Captchar wrote:Dan and Karen who live 10 miles apart meet at a cafe that is directly north of Dan's house and directly east of Karen's house. If the cafe is 2 miles closer to Dan's house than to Karen's house, how many miles is the cafe from Karen's house?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

ALWAYS LOOK FOR SPECIAL TRIANGLES.

The distance between Dan's house and Karen's house is 10.
Among the answer choices are 6 and 8.
The implication is a 6-8-10 triangle:

Since Karen lives 2 miles farther from the cafe than does Dan, the distance between her house and the cafe is 8.

The correct answer is C.

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