## Wednesday, August 15, 2007

### Must Pythagoream Theorem be used?

This question is found in the Yahoo!Answers as one of the question designed to test knowledge of the Pythagoream Theorem.

1. Sam hiked 48 miles directly east and then 36 miles directly south. Find the shortest distance in miles from the point where he ended up to his starting point.
A. 60 miles
B. 36 miles
C. 48 miles
D. 72 miles

For me, MCQ question should not be solved by calculation because answers are suggested and the little time allocated, it is designed not to be calculated.

If it is used for training, it is better to demand students to think and explain briefly on all the 4 suggested answers, instead of ignore them, just work out an answer and search to see which of the suggested answer matches it.

Here, obviously, B and C are wrong because from the question, Sam travelled 48 miles in one of the direction and both directions didn't make an angle of less than 90 degrees.

Thus, left with two possible answers A and D. Since 72 can be nicely expressed as 2x36, and if it is D, then probably you should have learnt another theorem about this nice feature. Thus, it is unlikely D.

So, assuming the MCQ is not faulty, and by considering the time and marks allocated ratio, it is wise to choose A and move on. When all is done, then can return to do checking with the left-over spare time. If the MCQ is faulty, it doesn't waste your time too since none will get it correct :)

Now, for the purpose of learning, it is better not just to calculate, but to figure out any nice pattern if the distance were 72 miles. By Pythagoream Theorem,
a^2 + b^2 = (2a)^2 ==> b^2 = 3 x a^2 ==> b = sqrt(3) x a ~ 1.7a > 1.5a ==> 1.5a < b, but
1.5 x 36 = 54 > b, thus distance cannot be 72. Alternatively, similar can be derived in a circle when a chord has same length as the radius, the chord that joins the other two ends of this chord and of the diameter that touches of this chord, is of length sqrt(3) x radius. Lastly, just a quick note, when using the standard way, it is faster to use mental calculation: 36^2 + 48^2 = (12x3)^2 + (12x4)^2 = 12^2 (3^2 x 4^2) = 12^2(5^2) = 60^2.