The time has come to issue a warning: this entire series of caches will be archived between May 1st and October 31st, 2022.
Triangles
Because the “learning” is cumulative, you will need to do most of this series in sequence - in alphabetical order of the caches which are named after famous mathematicians.
Caches in the Series
The series consists of: Archimedes (GC3TT8C), Bernoulli (GC3V9MW), Cayley (GC3W898), Descartes (GC3X0W7), Euclid (GC3XGN0), Fermat (GC413V5), Gauss (GC4219Z), Heron (GC4VGBX), Ingham (GC4WC72), Jarnik (GC4Z9PK), Kepler (GC4ZMRG), Lovelace (GC50BN6), Merryman (GC51D34) and Newton (GC5AJ04).
Note re Location of Final
As in other conservation areas in what used to be Victoria County - now "Kawartha Lakes" - there are signs warning of possible bear activity. We believe that this is a matter of the city's covering itself; it does not imply that any bears have been seen in the area in recent months or even years. However, the possibility does exist!
Descartes the Mathematician
René Descartes, a scholar of early C17th France, became known as the outstanding intellectual of his day. His writings in philosophy earned him a position among the most influential thinkers of all time. He invented “analytic geometry” - the mathematics related to the “Cartesian plane” with the X and Y axes - and the use of algebraic equations to locate points and determine relationships thereon. This alone merits his reputation as the “Father of Modern Mathematics” although his other credits constitute a lengthy list. Indicative of the esteem in which he is held is the fact that the University of Waterloo in Waterloo, Ontario, Canada, annually awards to students entering the Faculty of Mathematics, over $500 000 in scholarships of varying amounts but all bearing the name “Descartes.”
Descartes the Geocache: The Background Information
Some Definitions Related to Right-Angled Triangles
Draw neatly a right-angled triangle with sides of different lengths. Call it △XYZ with Y at the right-angled vertex. Label the three sides correctly (they go opposite the angles at X, Y and Z respectively - check “Archimedes” if you’ve forgotten) as x, y and z. We will consider ∡XZY which we will refer to as ∡Z.
We know that y is the hypotenuse because it is opposite the right angle at vertex Y. We know that, with respect to ∡Z, z is the opposite side - that’s why we called it z. The third side, x, which, with the hypotenuse, helps to form ∡Z, and is, therefore, beside or adjacent to ∡Z, is called the adjacent side with respect to ∡Z. Please label these three sides accordingly. (By the way, if we were considering ∡X, then x would be the opposite side and z the adjacent side.)
Some Special Ratios Related to Right-Angled Triangles
With respect to our ∡Z, we could write the following side ratios:
z/y which is the same as opposite/hypotenuse which we could shorten to o/h:
x/y which is the same as adjacent/hypotenuse which we could shorten to a/h;
z/x which is the same as opposite/adjacent which we could shorten to o/a.
In a right-angled triangle, these ratios are significant and are defined as follows:
o/h is defined as the sine of the angle, usually shortened to sin;
a/h is defined as the cosine of the angle, usually shortened to cos;
o/a is defined as the tangent of the angle, usually shortened to tan.
You may recall from school a method of remembering these ratios: SOH-CAH-TOA. This is simply a short form of: “sine is opposite over hypotenuse, cosine is adjacent over hypotenuse; tangent is opposite over adjacent.” Anyway, in this geocaching business, we don’t have to memorize this stuff (“when will we ever use it anyway?”) - just use it!! And, as we will see in the next cache in this series, this material can be really helpful in many practical ways - not just as a mathematical enigma! So hang in there!
What Happens if the Triangle is a Different Size?
We can hear you asking this question. The answer is that for every specific angle, these ratios are exactly the same regardless of the size of the angle. And the reason is that if the angles are the same, the triangles are similar and we learned in “Cayley” that if the triangles are similar, the ratios of corresponding sides are proportional - i.e. the ratios have the same value.
Illustration
Draw another triangle roughly identical to the one you drew before - with the same letters at the same vertices - but a new one because the first one is now - or should be! - cluttered. Make x the shorter of the two legs - about half the length of z - and mark the sides with these lengths: x = 8, z = 15 and y = 17 (the units are irrelevant but would obviously be the same).
Now state in fraction form:
(a) sin ∡X:
(b) cos ∡X:
(c) tan ∡X:
(d) sin ∡Z:
(e) cos ∡ Z:
(f) tan ∡ Z:
Answers: 8/17, 15/17, 8/15, 15/17, 8/17, 15/8.
The “Puzzles”
Note: In these puzzles, the letters for the answers - A, B, D, E, etc. - may be either one or two digits long. Before you draw each triangle, look at the side lengths given and make your triangle resemble that shape roughly - no scale drawing necessary.
1. Draw a right-angled triangle, △QRU, with q = 4, r = 5 and u = 3. (Remember, the longest side must be the hypotenuse!) In fraction form what is:
(a) sin ∡U?: _____; (B/D)
(b) cos ∡U?: _____; (E/F)
(c) tan ∡U?: _____; (G/I).
2. Draw a right-angled triangle, △AVW, with a = 13, v = 5, w = 12. In fraction form what is:
(a) sin ∡W?: _____; (J/K)
(b) cos ∡W?:_____; (L/M)
(c) tan ∡V?: _____ ; (N/P).
The Co-ordinates of the Cache
The co-ordinates are: N 44° ab.cde’ and W 078° fg.hij where the lower case letters are unrelated to any letters previously mentioned on this cache page. In addition, A is the tenths digit of the minutes part of the north co-ordinate of the location of the cache in “Cayley.”
To summarize:
A = ___; B = ___; D = ___; E = ___; F = ___;
G = ___; I = ___; J = ___; K = ___; L = ___;
M = ___; N = ___; P = ___.
a = 3(N) - M: ____
b = J - 2(I): ____
c = P - F: ____
d = 2(D) - G: ____
e = (P+ E) - (A + D): ____
f = 2[(L + M + N) - (B + F + K)]: ____
g = 4(B) - (E + G): ____
h = K - I: ____
i = A + I - (L + E): ____
j = J - E: ____.
The co-ordinates of the cache are: : N 44° __ __ . __ __ __ ’ and W 078°__ __ . __ __ __ ‘.
Other Notes
- Co-ordinates in here are "swingy:" sorry 'bout that!
- Please provide your own pen or pencil.
- The cache container is a clean camouflaged peanut butter jar - either a small regular or a large small, take your choice! - but, in spite of its size, it has some good "swag."
- The cache is located in a very pleasant place to walk, bicycle or ski. However, you should be aware that the minimum walking distance is about 530 m and, in the winter, about 1 km as the crow flies and, unfortunately, you won’t be able to walk in that straight a line!
- There is some bushwacking involved towards the end of the search.
Check Your Answer
You can check your answers for this puzzle on GeoChecker.com.