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705-805 (Hard)|   Geometry|      
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CapnSal
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A certain city has a circular wall around it and this wall has four 13 Apr 2023, 11:49
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Difficulty:

85% (hard)

Question Stats:

19% (02:11) correct 81% (02:29) wrong based on 21 sessions

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A certain city has a circular wall around it and this wall has four gates pointing North, South, East and West. A house stands outside the city, three miles North of the North gate and it can just be seen from a point nine miles East of the South gate. What is the diameter of the wall that surrounds the city?

(a) 6 miles
(b) 9 miles
(c) 12 miles
(d) 15 miles
(e) 20miles
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B
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Kinshook
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A certain city has a circular wall around it and this wall has four 13 Apr 2023, 13:01
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Given: A certain city has a circular wall around it and this wall has four gates pointing North, South, East and West. A house stands outside the city, three miles North of the North gate and it can just be seen from a point nine miles East of the South gate.

Asked: What is the diameter of the wall that surrounds the city?

Attachment:
Screenshot 2023-04-14 at 1.27.56 AM.png
Screenshot 2023-04-14 at 1.27.56 AM.png [ 34.81 KiB | Viewed 7279 times ]

3 (3 + 2r ) = D^2

(3+2r)^2 + 9^2 = (D+9)^2

D = 6 &
Diameter = 2r = 9
satisfy the equations

IMO B
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A certain city has a circular wall around it and this wall has four 13 Apr 2023, 14:28
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Can you please the equations you wrote and how you got them?

3 (3 + 2r ) = D^2
(3+2r)^2 + 9^2 = (D+9)^2

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A certain city has a circular wall around it and this wall has four 14 Apr 2023, 08:09
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CapnSal
A certain city has a circular wall around it and this wall has four gates pointing North, South, East and West. A house stands outside the city, three miles North of the North gate and it can just be seen from a point nine miles East of the South gate. What is the diameter of the wall that surrounds the city?

(a) 6 miles
(b) 9 miles
(c) 12 miles
(d) 15 miles
(e) 20miles
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Three ways.

Knowing How Test Writers Craft Questions
The base of the triangle is 9 and all the numbers in the question and answer choices are whole numbers. We are almost certainly looking at a special right triangle with integer value sides. Could it be a 3-4-5? If so, the base of 9 will correspond to the 3, making the height 12. Does that look like it makes sense with the rest of the figure? Yep. Done. I'm 100% serious that this is how I would do this question on an actual test. You get no points for doing "real" math if you can just anticipate how test questions are written.
Answer choice B.

Plugging In The Answers (PITA)
Refer to Kinshook's diagram (so I don't have to redraw it :) ).
A) If the diameter is 6, the height is equal to the base. Is that true? No.
B) Height = 12, base = 9, hypotenuse = 15, so the segment labeled D is 6. Does that seem reasonable? Yes, so keep it.
C) Height = 15, base = 9, hypotenuse = 17.5, so the segment labeled D is 8.5. Does that seem reasonable? No, so eliminate C and anything larger.
Answer choice B.

Estimation
As with the vast majority of geometry questions, eyeballing and estimation go a LONG way. Draw the figure and look at it. Really LOOK at it. The segment that Kinshook labeled as D - how long is it?
Well, it's definitely longer than 3. I'd feel good about estimating it to definitely be at least 5.
Okay, it's also definitely shorter than 9. I'd feel good about estimating it to be definitely be at most 7.
I'd just roll with 6 and be on my merry way with a 9-x-15 triangle, which I can see is a juiced up 3-4-5, making the vertical leg equal to 12, so the diameter equal to 9.
But if you're less confident in rolling with 6 and need to prove to yourself, you can Pythagorean using hypotenuse equals 14 and hypotenuse equals 16. Those result in a lower bound for the diameter of 7.7 and an upper bound of 10.2.
Answer choice B.
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