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## Question

Allen and Bernard decide to go visit their friend Chuck. Bernard lives on the road that runs due East of Allen and Chuck lives on the road that runs due North of Allen. Bernard doesn’t know how far it is to Chuck’s house but he does know that their friend Edward lives on the road that runs from Bernard’s house to Chuck’s and it is 1 mile to Edward’s house. Allen doesn’t know how far any of his friend’s houses are from his, but he does know that a stream runs from his house to Edward’s and it intersects the road at a right angle.

Bernard sets out from his house and heads toward Allen’s and a little while later Allen sets out from his house headed toward Bernard’s. When they meet they decide to cut straight through the woods to Chuck’s house crossing over the stream. When Bernard meets Allen he has covered half of distance he will travel to Chuck’s house, and he has gone equal the distance from his house to Edward’s.

When Allen leaves Chuck’s house, he heads straight home. How far did Allen travel to and from Chuck’s house?

All roads and streams and paths through the woods are straight lines.

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Allen and Bernard decide to go visit their friend Chuck. Bernard lives on the road that runs due East of Allen and Chuck lives on the road that runs due North of Allen. Bernard doesn’t know how far it is to Chuck’s house but he does know that their friend Edward lives on the road that runs from Bernard’s house to Chuck’s and it is 1 mile to Edward’s house. Allen doesn’t know how far any of his friend’s houses are from his, but he does know that a stream runs from his house to Edward’s and it intersects the road at a right angle.

Bernard sets out from his house and heads toward Allen’s and a little while later Allen sets out from his house headed toward Bernard’s. When they meet they decide to cut straight through the woods to Chuck’s house crossing over the stream. When Bernard meets Allen he has covered half of distance he will travel to Chuck’s house, and he has gone equal the distance from his house to Edward’s.

When Allen leaves Chuck’s house, he heads straight home. How far did Allen travel to and from Chuck’s house?

All roads and streams and paths through the woods are straight lines.

2.5 miles? .4 toward Bernard's house, 1 mile to Chuck's house, 1.1 miles back home....

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3.017 miles = 0.327 + 1.365 + 1.325.

Allen lives 1.325 miles from Chuck; Bernard lives 2.150 miles from Chuck; Allen & Bernard live 1.692 miles from each other. Edward lives 1 mile from Allen (given), 1.365 miles from Bernard, and 0.784 miles from Chuck. The point where Allen & Bernard meet is 0.327 miles from Allen's house and 1.365 miles from Bernard's & Chuck's houses. I'm sure there's other pertinent information, but that's enough for me.

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Quick clarification request.

When Bernard meets Allen he has covered half of distance he will travel to Chuck’s house, and he has gone equal the distance from his house to Edward’s.

When they meet, do you mean Allen has covered half the distance he will travel or Bernard has covered half the distance? And from "his house to Edwards," does this refer to Allen or Bernard?

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Quick clarification request.

When they meet, do you mean Allen has covered half the distance he will travel or Bernard has covered half the distance? And from "his house to Edwards," does this refer to Allen or Bernard?

My assumption was that both referred to Bernard, as it appears Bernard is the subject of the initial clause and Allen is the object. I think good grammar would keep Bernard as the subject throughout. Although I hate relying on grammar to figure out a geometry problem...
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Quick clarification request.

When they meet, do you mean Allen has covered half the distance he will travel or Bernard has covered half the distance? And from "his house to Edwards," does this refer to Allen or Bernard?

My assumption was that both referred to Bernard, as it appears Bernard is the subject of the initial clause and Allen is the object. I think good grammar would keep Bernard as the subject throughout. Although I hate relying on grammar to figure out a geometry problem...

Yes. I was referring to Bernard in both cases. When Bernard meets up with Allen he is halfway through his trek to Chuck's house. The length of his trip can be determined as twice the distance from Bernard's to Edward's.

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3.017 miles = 0.327 + 1.365 + 1.325.

Allen lives 1.325 miles from Chuck; Bernard lives 2.150 miles from Chuck; Allen & Bernard live 1.692 miles from each other. Edward lives 1 mile from Allen (given), 1.365 miles from Bernard, and 0.784 miles from Chuck. The point where Allen & Bernard meet is 0.327 miles from Allen's house and 1.365 miles from Bernard's & Chuck's houses. I'm sure there's other pertinent information, but that's enough for me.

But we know that Bernard has travelled only 1 mile (equal the distance from his house to Edward's) when he meets Allen.

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But we know that Bernard has travelled only 1 mile (equal the distance from his house to Edward's) when he meets Allen.

I had 1 mile from Allen to Edward, not Bernard to Edward

So that means the distances from Bernard to Edward, Bernard to the point where he meets Allen, and from that meeting point to Chuck are all 1 mile. You'd think that would make the equations easier, but it didn't. Anywhooo... now I get:

Allen walks 0.579 + 1 + 0.815 = 2.394 miles.

Allen lives 0.815 miles from Chuck; Bernard lives 1.414 miles from Chuck; Allen & Bernard live 1.579 miles from each other. Edward lives 1 mile from Bernard (given), 1.222 miles from Allen, and 0.414 miles from Chuck. The point where Allen & Bernard meet is 0.579 miles from Allen's house and 1 mile from Bernard's & Chuck's houses.

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I had 1 mile from Allen to Edward, not Bernard to Edward

So that means the distances from Bernard to Edward, Bernard to the point where he meets Allen, and from that meeting point to Chuck are all 1 mile. You'd think that would make the equations easier, but it didn't. Anywhooo... now I get:

Allen walks 0.579 + 1 + 0.815 = 2.394 miles.

Allen lives 0.815 miles from Chuck; Bernard lives 1.414 miles from Chuck; Allen & Bernard live 1.579 miles from each other. Edward lives 1 mile from Bernard (given), 1.222 miles from Allen, and 0.414 miles from Chuck. The point where Allen & Bernard meet is 0.579 miles from Allen's house and 1 mile from Bernard's & Chuck's houses.

Looking at the triangle formed from Allen's, Edward's, and Chuck's house (triangle ACE), we know that a line (the stream) runs from Allen's to Edward's and is perpendicular to the road at Edward's house. So ACE is a right triangle with AC as the hypotnuese. So (Edward to Chuck)2 + (Allen to Edward)2 = (Allen to Chuck)2. But .4142 + 1.2222 does not equal .8152.

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2.88784 miles. ABC is an equilateral right triangle, which is bisected by E. BE=CE=1, and BC=2. Chuck's house is sqrt(2) due North of Allen. Allen travels sqrt(2)-1 miles east before meeting Bernard. Then they travel sqrt((sqrt(2)-1)^2+sqrt(2)^2) NorthEast to Chucks, and then Allen returns home.

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2.88784 miles. ABC is an equilateral right triangle, which is bisected by E. BE=CE=1, and BC=2. Chuck's house is sqrt(2) due North of Allen. Allen travels sqrt(2)-1 miles east before meeting Bernard. Then they travel sqrt((sqrt(2)-1)^2+sqrt(2)^2) NorthEast to Chucks, and then Allen returns home.

1.474. But we know that after Allen and Bernand meet they continue together to chuck's house and travel 1 mile (Bernard travels twice the distance from B to E and he has gone half-way when he meets Allen). If from where they meet to Chuck's is 1 mile, it follows that from Allen's to Chuck's should be less then 1 mile, since the path they both take after meeting is the hypotnuese of another right triangle and Allen to Chuck's is the adjacent side and Allen's to the meeting point is the oppisite side.

Spoiler for I'll lend a hand:

Prove that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they cross the stream.
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When Bernard meets Allen he has covered half of distance he will travel to Chuck's house, and he has gone equal the distance from his house to Edward's.

2.88784 miles. ABC is an equilateral right triangle, which is bisected by E. BE=CE=1, and BC=2. Chuck's house is sqrt(2) due North of Allen. Allen travels sqrt(2)-1 miles east before meeting Bernard. Then they travel sqrt((sqrt(2)-1)^2+sqrt(2)^2) NorthEast to Chucks, and then Allen returns home.

I think you have the distance between Bernard's house and the meeting point as 1 mile, but the distance between that meeting point and Chuck's house is larger than 1 mile, which doesn't agree with the above point from the OP.

Bernard should travel 2 miles to Chuck's, and Allen, because he leaves his house after Bernard leaves his, should travel more than 1 mile but less than 2 to get to Chuck's. I don't read that CE=1 (necessarily) from the OP. If you place point D as the meeting point for Allen and Bernard, I read that BD = CD = BE = 1.

After cranking the numbers some more, I now think that Allen walks 0.260 + 1.000 + 0.965 = 2.225 miles.

Allen lives 0.965 miles from Chuck; Bernard lives 1.587 miles from Chuck; Allen & Bernard live 1.260 miles from each other. Edward lives 1 mile from Bernard (given), 0.766 miles from Allen, and 0.587 miles from Chuck. The point where Allen & Bernard meet is 0.260 miles from Allen's house and 1 mile from Bernard's & Chuck's houses.

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Prove that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they cross the stream.

I get that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they meet, both distances being 0.26 miles. And here's a diagram with my solution:

Edited by HoustonHokie
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I get that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they meet, both distances being 0.26 miles. And here's a diagram with my solution:

Prove that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they meet. My bad.

Those distances are spot on!

Edited by Prof. Templeton
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I get that the distance from where Allen and Bernard meet to where they cross the stream is the same as the distance from Allen's house to where they meet, both distances being 0.26 miles. And here's a diagram with my solution:

Oh, I see...I was reading "When Bernard meets Allen he has covered half of distance he will travel to Chuck's house, and he has gone equal the distance from his house to Edward's." as the distance he would if he was going straight to Chuck's house (along BC). I guess I read "will" as "would"...since he was heading towards Allen's house, not Chuck's...oh well, word problems were almost my worst sub-subject of math...

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Anyone care to do a proof to show why x = (3√2)-1 ? I prefer the geometric one, but to each his own.

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