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# Dan and Karen, who live 10 miles apart meet at a cafe that

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Manager
Status: May The Force Be With Me (D-DAY 15 May 2012)
Joined: 06 Jan 2012
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Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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31 Mar 2012, 20:07
2
6
00:00

Difficulty:

35% (medium)

Question Stats:

76% (02:07) correct 24% (02:08) wrong based on 253 sessions

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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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Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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01 Apr 2012, 01:40
1
boomtangboy wrote:
14. 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

Look at the diagram below:

As you can see we have a right triangle with the sides of x, x-2 and hypotenuse of 10. If you can not recognize on sight that this triangel must be 6-8-10 right triangle, you can use Pythagoras theorem and find x: x^2+(x-2)^2=10^2 --> x=8.

Attachment:

The distance.png [ 4.31 KiB | Viewed 10737 times ]

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Re: Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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08 Aug 2012, 10:09
Its a question on Pythagoras triangles. the most familiar is 5-4-3. Here also if u notice, its a multiple of basic Pythagoras triangle 2(5-4-3) or 10-8-6.
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Dan and Karen, who live 10 miles apart, meet at a cafe that is  [#permalink]

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24 Aug 2017, 21:17
Since the cafe is directly east to Karen's house and directly north to Dan's house,
they form the two legs of a right angled triangle. From the question stem,
If we assume that the distance between Karen's house and the cafe is x,
then, distance between Dan's house and the cafe is x-2

Also, the distance between their houses is the hypotenuse of the triangle(which is 10)

In a right triangle, $$Hyp^2 = Side1^2 + Side2^2$$
Substituting the values, $$10^2 = x^2 + (x-2)^2$$
6,8,10 is a Pythagorean triplet which satisfies this equation

Therefore, the distance from Karen's house to the cafe is 8 miles(Option C)
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Dan and Karen, who live 10 miles apart, meet at a cafe that is  [#permalink]

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24 Aug 2017, 21:18
Bunuel 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

From the figure it is evident that
$$10^2 = x^2 +(x-2)^2$$ solving this we will get

$$x^2 - 4x + 4 = 100$$ or $$x^2 - 2x - 48 = 0$$

Hence $$x = 8$$

option C
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Posts: 7334
Re: Dan and Karen, who live 10 miles apart, meet at a cafe that is  [#permalink]

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24 Aug 2017, 22:02
Bunuel 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

Hi..

ofcourse one method is the algebraic as explained above..

the other is to realize that MORE than often, you may be tested for triplet is 3:4:5.. or may be 1:$$\sqrt{3}$$:2, a 30-60-90 triangle or 1:1:$$\sqrt{2}$$, a 45-45-90 triangle
and so when you see the HYPOTENUSE here as 10, a multiple of 5, try out this triplet ( may save some time)

so 3:4:5 has 5 as hypotenuse and with 10 as hypotenuse it becomes 3*2:4*2:5*2 = 6:8:10..
so Karens distance is TWO more and 6:8 perfectly fit in..
ans 8
C
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Re: Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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08 Oct 2018, 09:38
1
Hey guys the simplest approach is to plug in the answer choices and you immediately see that the correct answer has to be C.
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Re: Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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09 Oct 2018, 20:08
Hi,

Could someone explain with an algebraic equation as to how the distance between Dan's house and the cafe is x-2?

Thanks
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Re: Dan and Karen, who live 10 miles apart meet at a cafe that  [#permalink]

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10 Oct 2018, 17:48
boomtangboy 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

We see that the three locations (Dan’s house, Karen’s house and the cafe) will form a 6-8-10 right triangle. That is, Dan’s and Karen’s houses are 10 miles apart (which is given) and the cafe is 6 miles from Dan’s house and 8 miles from Karen’s house (notice that this makes the cafe 2 miles closer to Dan’s house than to Karen’s house). Therefore, the cafe is 8 miles from Karen’s house.

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Re: Dan and Karen, who live 10 miles apart meet at a cafe that   [#permalink] 10 Oct 2018, 17:48
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# Dan and Karen, who live 10 miles apart meet at a cafe that

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