Distance between towns problem w/ only 1 angle given

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geekily

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"A drive from Princeton to Queenstown currently requires passing through Regalville, due to a steep hill between Princetown and Queenstown. The two existing roads are level and straight, they meet at Regalville at a 100º angle. Make an accurate scale drawing of the sketch and use it to figure out how many miles would be saved driving from Princeton to Queenstown if engineers created a tunnel through the hill and a direct road from Princeton to Queenstown was created." (It shows a sketch of angle PRQ, with angle R being 110º. Side PR is 12 miles, and RQ is 17 miles.)

We did a problem similar to this in class once, but we were given 2 angle measures - so when we drew the 2 angles out until they met, we drew a line straight down to get the side we needed. I'm not quite sure how to approach this with only one angle. I tried redrawing a 110º angle and actually measuring the sides of the sketch in cm to get the right proportion, even though in class she said it didn't matter what size our new sketch was - so then I tried just making whatever length I wanted with a 110º. She said not to assume pictures are to scale unless it explicitly said so, so I didn't try measuring the other angles. Either way, I divided the 17 or 12 miles with whatever I replaced them with (say, 17 miles/6 cm) and set that equal to x mi/6.8 cm, the length of PQ in my new sketch.) Then I cross-multiplied. She said not to worry about units because as long as they matched (miles on the top and cm on the bottom, it would cancel out.) I subtracted my answer from 29 (12+17) to find the difference, which would be the distance saved. The correct answer is 5.2 mi, but whatever I did, I couldn't get close to that. What am I doing wrong?

Thanks for your help. :)
 
1) Feel free to use equations and symbolic expressions. We do that here in the world o'mathematics.

2) Don't "cross multiply". Very ugly. Learn to multiply.

3) Have you the Law of Cosines?
 
Hmpf... Well, then I would have to suggest that you measure very carefully. You will have to decide if it is 100º or 110º. You've written both.

Connect P and Q. Now you've three sides. I'm not seeing any similar triangles in there, so I'm not sure how that proportion will help you. You can be sure PQ is greater than 17 and less than 29. That might be enough help to draw a good scale drawing.
 
Oh, I'm sorry for the typo - it's a 110º angle. I'll try your suggestion. Thanks!
 
I've measured as carefully as I can, and made several different attempts. I can't get anywhere close.
 
A drive from Princeton to Queenstown currently requires passing through Regalville, due to a steep hill between Princetown and Queenstown. The two existing roads are level and straight, they meet at Regalville at a 100º angle. Make an accurate scale drawing of the sketch and use it to figure out how many miles would be saved driving from Princeton to Queenstown if engineers created a tunnel through the hill and a direct road from Princeton to Queenstown was created." (It shows a sketch of angle PRQ, with angle R being 110º. Side PR is 12 miles, and RQ is 17 miles.)

We did a problem similar to this in class once, but we were given 2 angle measures - so when we drew the 2 angles out until they met, we drew a line straight down to get the side we needed. I'm not quite sure how to approach this with only one angle. I tried redrawing a 110º angle and actually measuring the sides of the sketch in cm to get the right proportion, even though in class she said it didn't matter what size our new sketch was - so then I tried just making whatever length I wanted with a 110º. She said not to assume pictures are to scale unless it explicitly said so, so I didn't try measuring the other angles. Either way, I divided the 17 or 12 miles with whatever I replaced them with (say, 17 miles/6 cm) and set that equal to x mi/6.8 cm, the length of PQ in my new sketch.) Then I cross-multiplied. She said not to worry about units because as long as they matched (miles on the top and cm on the bottom, it would cancel out.) I subtracted my answer from 29 (12+17) to find the difference, which would be the distance saved. The correct answer is 5.2 mi, but whatever I did, I couldn't get close to that. What am I doing wrong?

The correct answer could not possibly be 5.2 miles. WIth angle PRQ being 110º, PQ has to be greater than 17 for sure, if your picture is correct.

Another approach to determining PQ is as follows:

1--Draw a horizontal line, RQ, 17 umits long, R at the left end.
2--Draw another line 12 units long, upward, and to the left, making an angle of 110º to RQ, the end poing being P.
3--Drop a vertical line from P to RQ extended to the left, meeting at point A.
4--Angle RPA = 20º.
5--AR = 12sin(20) = 4.1042
6.AQ = 21.1042
7--PA = 12
8--Therefore, PQ = sqrt(12^2 + 21.1042^2) = 24.277 miles.
9--I made a sketch with RQ = 6cm (17 miles), constructed line PQ at 110º to RQ measuring 4.235cm (12 miles), and measured PQ to be ~8.4cm making PQ = ~24 miles.

I suggest you examine your figure which should look like the following:

P
.*
..\*
...\..*
....\....*
.....\......*
......\........*
.......\..........*
........\............*
.........\..............*..Clearly not to scale
..........\................*..RQ = 17, PR = 12
...........\..................*
............\....................*
.............\......................*
..............\........................*
...............R.......................Q

CLearly, your figure is wrong as PQ could not possible measure 6.8cm if RQ = 17 miles measures 6cm.
 
Wow, thank you so, so much! Just by looking at your step 9 I was able to see my error - I wasn't doing 17 mi/6 cm = 12 mi/x cm and therefore PR was not proportional. What amazes me is that you measured it perfectly, because using your numbers, PQ = 23.8, and 29 - 23.8 = 5.2, which was the given answer. I drew 3 different figures and I was never able to come closer than PQ = 8.5 cm, so I got 24.084, and an ending answer of 4.916. Guess my pictures need some work, heh. But still, she said she would allow a margin of error of 10% on the test, so I think I'm close enough. At any rate, now I know what to do! Thank you so much!

Hmm, actually, it's 5% in either direction margin of error, so I'm not quite there. My answer's .284 off and allowable is .26. Gaaah, so close!
 
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