Maths problem - you are told that one side of a 'right triangle' (as the Americans call them), is a certain fraction (1/n) of the total perimeter (the length of all three sides of the triangle added together). You have to work out the length of the other sides and/or the area.
For example... you are told that one side (the height) is one-sixth of the total perimeter.
Write down "n = 6" and also 'adjusted n' (for which I use capital "N").
N = n-1, so write down "N=5".
The relative length of the height is N x 2 'units'. Note: 'units' means relative length, not absolute length.
If n = 6 then N = 5 and height N x 2 = 10 units.
The total length of the other two sides = height (10) x N (5) = 50 units.
If this were an isosceles triangle, the other two sides would simply be half that, 25 units each.
In a right triangle, the hypotenuse is longer than the base, and in these problems, the hypotenuse is simply 2 units longer than the base (don't ask me why, that's just how this works).
So take the side length from the theoretical isosceles triangle (25 units), add 1 for hypotenuse (= 26 units) and deduct 1 for base (= 24 units).
That gives you height 10 units, base 24 units, hypotenuse 26 units, total perimeter 60 units.
(which you could simplify to 5-12-13 if you wished)
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You might be told, or able to work out, that the total perimeter is (say) 72 centimetres (or 90 yards), or whatever, you divide that by the number of units (= 1.2 centimetres/unit or 1.5 yards/unit) and multiply up again.
So sides would be height 12 cm; base 28.8 cm, hypotenuse 31.2 cm (or 15 yards; 36 yards; 39 yards).
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Area is simply base x height x 1/2.
So using above examples, area would be 12 x 28.8 x 1/2 = 172.8 sq cm (or 270 sq yards).
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Presh Talwalkar, at the end of this video, reckons you can short-circuit calculating the area if you know the perimeter and "n".
The formula, if you can remember it, is...
(perimeter ^2) x (N-1) ÷ (4N^2 + 4N).
So if n = 6, N = 5 and perimeter is 72 centimetres (or 90 yards), the area is...
72 x 72 x 4 ÷ (100 + 20) = 172.8 sq cm.
90 x 90 x 4 ÷ (100 + 20) = 270 sq yards.
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Here endeth.
Debate Magazine
