Monday, 17 February 2020

Fun with numbers - right-angle triangles and side length ratios.

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" to get started.

The term "n" is hardly used from here on in, what comes up all the time is "n-1" so we might as well treat this as a separate variable, 'modified n', for which I use capital "N".

So write down "N = 5".

The relative length of the height is simply N x 2* 'units'**.

* Note: override rule: if "N" is an odd number, see separate section below. But I think it's easier to just remember one rule which works whether "N" is odd or even.

** Note: 'units' means relative length, not absolute length.

So write down 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. That's the magic here. I'll have to have a think about why, but for now, just accept that that's how this works.

(UPDATE, I've had a think and explained it (to myself, at least) and it turns out to be pretty non-magical at all. The height is always 2N units and the difference between base and hypotenuse is always units because that's how you set up the equations. Still a handy trick, should you ever need it.)

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 can simplify to 5-12-13. If you apply the override rule, you would have got 5-12-13 straightaway.
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Override rule example, same facts as above:

n = 6, N = 5, so "N" is odd.

Height = N = 5 units.

Total length of other two sides = height x N (or N^2, if you are that way inclined) = 25 units.

In an isosceles triangle, the other two sides would be 12.5 each. This is a right triangle, so add and deduct 0.5 to find length of hypotenuse and base.

Answer = 5 - 12 - 13.

Remember, this only works if "N" is odd; the rule that height = N x 2 always works, whether "N" is odd or even!
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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.

4 comments:

James Higham said...

Wot, before breakfast?

Mark Wadsworth said...

JH I couldn't sleep very well, so I did the calcs in my head and then looked for a pattern.

Bayard said...

"The relative length of the height is simply N x 2* 'units'**."

I can't see from where you derive that.

Mark Wadsworth said...

B, quadratic equations, which I did in my head. I might do a follow up post on this.