|
Here's a puzzle for you. Read carefully and draw a good picture.
Lost Lake Road lies 300 yards north of Vanda's house and runs straight east and west. Art lives 600 yards south of Lost Lake Road and 1200 yards down the road from Vanda. Neither Art nor Vanda has a phone. Art's carrier pigeon just arrived at Vanda's screeching loudly with an emergency message from Art. He has fallen and broken his leg at home. Vanda decides to ride her dirt bike from her home to Lost Lake Road to try to flag down a passing motorist with a cell phone so that an ambulance can be sent to pick up Art. Vanda wants to ride in a straight line to Lost Lake Road, and then go straight from that point to Art's house. Where must Vanda hit Lost Lake Road (in the 1200 yard stretch) to make the distance she travels the shortest? Remember that you must explain how you got the answer and you must convince me that you are right. SolutionsThis problem proved to be pretty difficult - a lot of people managed the right numbers, but only 37 people could actually convince me that their answer led to the shortest path that Vanda could travel. 90 people got it wrong. The focus here is on explaining why it's the shortest. Reasons such as, "It makes similar triangles" or "they're Pythagorean triples" are not acceptable. Those are true statements, but don't really explain why that's the spot we want. There are at least three "reliable" ways to tackle this problem. The most basic is to know that the shortest distance between two points is a straight line. So how do we turn this problem into something that most closely resembles a straight line? The best way, as I see it, is to reflect one of the houses over the road, and then connect them with a straight line. Matt Smawfield of the American Embassy School in New Delhi and Sonya Kharas of Georgetown Day School did it this way and included some super pictures, so I've given their answers below. Take a look and see how it works if you are not familiar with this. Another version on this same theme is what Jenny Kaplan did. She mentioned a physics fact that some of you know, and others of you will learn eventually, that light travels the shortest possible distance, and when you bounce it off something, the angle that it bounces off at is the same as the angle that it hits at. This can be proved by using the reflection, as above. You can read her solution below as well. A third way to do it is to find a way to minimize the hypotenuses of those two triangles. One way to do this is to graph the equation on your calculator and ask the calculator what the minimum value is. This isn't my favorite, but it works, and you still have to know what you're trying to find. Zach Rentz of Akiba Hebrew Academy was one person who did it that way, and his solution is highlighted below. The second more 'advanced' way to approach this same method is to find the same equation, but then to use calculus to find the minimum of the equation (that means actually doing the calculation that the calculator did in the example above). Both Lim Yim of Raffles Institution and Jason Chiu of Laramie Junior High School did it this way, and you can take a look at their solutions below to see what you have to look forward to when you take calculus. A number of people took some spots along Lost Lake Road and calculated the total distance Vanda would have to travel for each one, and based their conclusions on that. That's a good start, but rarely did I give credit for this. Only if they talked of seeing the pattern of lengths and checking very carefully around the spot did I consider credit. How do you know, if you check every 300 yards, or even 100 yards, that it's not 10 yards away from the spot you picked? A few people misinterpreted the picture, which was a possibility based on some tests I did here in the office (I asked a few people to draw me a picture of the problem), but they got credit as long as they described what they were doing and still got the right answer using good, reliable methods. I got a number of answers that stated that the minimum distance traveled was less than 900 yards. First off, the problem asks not for the minimum distance, but for a spot on the road. Second, how can it be less than 900? Their driveways alone add up to 900, and we haven't considered how to get from one to the other yet. Read carefully, and always check to see if your answer makes sense when you're done. A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.
From: Lim Yin
Grade: 10
School: Raffles Institution, Singapore
I don't have any graphic stuff, hope this "drawing"'s good enuff!
M X N
============..=================================
| .. .. |
| .. .. |
|... .. |
V .. |
.. |
...|
A
The dots represent Vanda' chosen path.
I've got two solutions from the diagram.
-----
First, let the length of MX be p. The formula for the length of the line would
be
[square root of ((p*p)+(300*300))] +
[square root of ((1200-p)*(1200-p)+(600*600))]
We want to find the minimum value of the expression above for 0 |
[Privacy Policy] [Terms of Use]
© 1994-2009 The Math Forum
Contact Us
I loved this problem. First, I needed to remember that the shortest distance
between any two points is a straight line. I then drew point H, which was
equidistant from line AB as point C (Vanda's House.) Next I drew segment HD
which would be the shortest distance. Segment HD intersected line Ab at point
I. To prove that segment HI was congruent to CI, therefore showing that
segment HD was congruent to the sum of the measures of segments CI and ID, I
drew the triangles HEI and CEI. I knew that segment CE was congruent to
segment EH (both 300 yards), and that angle HEi was congruent to angle CEI (all
right angles are congruent) and that segment EI is congruent to segment EI
(reflexive property.) I now had S.A.S. is congruent to S.A.S. and I could
prove that the two triangles are congruent. segment CI and segment HI could
now be proved congruent because they are corresponding parts of congruent
triangles. Now that I knew that point I was where Vanda had to reach Lost Lake
Road, how could I find out exactly where that point was? Easy. First I proved
angle HIE congruent to angle FID (vertical angles are congruent) and I proved
that angle HIE was congruent to angle CIE (corresponding parts of congruent
triangles are congruent.) By the transitive property I now knew that angle EIC
was congruent to angle FID. I knew these two angles were congruent and that
angle IEC was congruent to angle IFD (all right angles are congruent,) and I
knew that triangle CEI was similar to triangle DFI. I labeled segment EI (x)
and segment IF (y). I had two equations with these two variables. First I
knew that x + Y =1200 yards, and I knew that x/y = 1/2 (since in similar
triangles the ratio of sides is congruent and the ratio of segment EC to
segment FD is 300/600 or 1/2. I therefore solved the two equations and found
that x=400 yards and y =800 yards. I knew now that point I was 400 yards from
point A and that Vanda had to pass through this point to find the shortest
distance to Art's house.
From,
Sonya Kharas
To have a basic idea of similar triangles, you have to use a little
imagination to picture that there is a mirror across Lost Lake Road. If
there were a mirror on the edge of a pool table, then if you aimed for
the pocket by looking at the reflection on the mirror placed on the bank,
the ball would bounce off the bank and go straight into the hole. This
idea is applied in this problem. If you extended the 300yd line from
Vanda's house to Lost Lake Road another 300 yd, and drew a line from this
mirror point to Ark's house, you should get a perfectly straight line,
which intersects Lost Lake Road at a certain point. Then, draw a line
from Vanda's house, to the intersection. The diagram should then be
divided into two triangles. (see fig. 2) The special thing about these
triangles, is that they are similar. The 300yd and the 600yd streets are
parallel, and considering that the line intersects not only Lost Lake
Road, but these two parallel lines as well, it becomes evident that the
angles of these triangles are congruent. The angles where the streets
intersect the road are both perpendicular, therefore, these two interior
angles are congruent, and therefore completing the proof that these two
triangles are similar. Now that we know that these two triangles are
similar, we find the ratio of two sides. The two sides, which are
available, are the length of the streets, of 300yd and 600yd, which each
make up a leg of each triangle. The ratio is 300yd/600yd = 1:2. Now if
we apply this ratio to the other legs of the triangles, we can find at
which point Vanda should stop at the road. 1x + 2x = 1200yd, 400yd +
800yd = 1200yd, 1:2 = 400:800. This then gives us the answer to the
problem, which is where along Lost Lake Road should Vanda hit, to make
the distance she travelled the shortest. If she stops 400 meters along
the road, she will be on the shortest route.
There are many reasons, which prove this theory true. One way would be to
find the total length of the journey (the addition of the two
hypotenuses), and comparing it with other differences. Fortunately, the
triangles in this method are similar, and they are also 3.4.5 triangles,
therefore making it easy to find the hypotenuses. 500yd + 1000yd =
1500yd. (see fig. 3) No matter whatever distance you try, you will never
find a total distance, which is less than 1500yd. The other way of
proving this theory, is by taking another look at the mirror concept. In
this solution, the line is straight, and therefore it is the shortest. By
altering the distance from 400, 800, the straightness of the line will be
altered, and therefore the total distance will change, after all a
straight line is shorter than a curved or bent line. Thus, the shortest
distance possible would be if Vanda stopped 400 meters down Lost Lake
Road, no shorter, no further, giving the optimum amount of time and
distance efficiency.
Answer: To achieve the shortest travelling distance, Vanda has to hit
Lost Lake Road diagonally at the 400yd (from the West) or at the 800yd
(from the east) mark, and then go diagonally again to Art's house.