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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Anurag@Gurome
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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, x² + (x - 2)² = 10² = 100
--> x² + x² - 4x + 4 = 100
--> 2x² - 4x - 96 = 0
--> x² - 2x - 48 = 0
--> x² - 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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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.Anurag@Gurome wrote:So, x² + (x - 2)² = 10²
Hence, 6² + 8² = 10² and (8 - 6) = 2
Hence, the cafe is 8 miles from Karen's house.
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ALWAYS LOOK FOR SPECIAL TRIANGLES.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
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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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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