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# City B is 5 miles east of the city A, city C is 10 miles southeast of

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Math Expert
Joined: 02 Sep 2009
Posts: 61283
City B is 5 miles east of the city A, city C is 10 miles southeast of  [#permalink]

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11 Nov 2019, 00:50
1
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Difficulty:

95% (hard)

Question Stats:

24% (02:57) correct 76% (02:00) wrong based on 42 sessions

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City B is 5 miles east of the city A, city C is 10 miles southeast of city B. Which of the following is the closest to the distance from city A to city C?

A. 11 miles
B. 12 miles
C. 13 miles
D. 14 miles
E. 15 miles

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Joined: 07 May 2019
Posts: 210
Re: City B is 5 miles east of the city A, city C is 10 miles southeast of  [#permalink]

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10 Feb 2020, 01:09
Bunuel wrote:
City B is 5 miles east of the city A, city C is 10 miles southeast of city B. Which of the following is the closest to the distance from city A to city C?

A. 11 miles
B. 12 miles
C. 13 miles
D. 14 miles
E. 15 miles

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Solution:
• Here, the keyword is Southeast, which means DBC is exactly 45 degrees.

• In right-angle triangle DBC,
o $$∠DBC = ∠DCB = 45$$
o It means triangle DBC is an isosceles triangle, and $$BD = CD = x$$ (say)
$$10^2=x^2+x^2$$
$$x=\frac{10}{√2}$$
• Now, in right angle triangle ADC,
o $$AD = AB+BD=5 + \frac{10}{√2}$$
o $$CD=\frac{10}{√2}$$
o $$AC^2=(5+\frac{10}{√2})^2+(\frac{10}{√2})^2$$
$$AC^2=25+\frac{100}{√2}+\frac{100}{2}+\frac{100}{2}$$
$$AC^2=125+\frac{100}{√2}=125+70$$ $$\ \ \ \ \ \ \ [\frac{100}{√2}=50√2=70(approx.)]$$
$$AC=√195$$, which is nearest to $$14$$.
Hence, the correct answer is Option D.
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Re: City B is 5 miles east of the city A, city C is 10 miles southeast of   [#permalink] 10 Feb 2020, 01:09
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