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More triangles!

Earlier this evening, Jane Moreton (@PGCE_Maths) tweeted this:

image

I looked at it for a moment and started pondering. In this case comes was clearly always going to be 45. I wondered whether the others would change and then realised they wouldn’t. I decided to draw it out and play around with some angle reasoning.

When I drew it out something else occured to me, there would be 3 right angled triangles, all with the same opposite side (from the given angle) and differing adjacent sides that were multiples of each other. It occurred that the sidelengths didn’t matter, and I could reason it out with tangent ratios instead (and everyone knows trigonometry is more fun).

Tan (a) = 1/3
Tan (b) = 1/2
Tan (c) = 1

I realise that as the angles will always be the same I could evaluate each one and show that a+b = c. But a) I didn’t have a calculator on me and b) that’s no fun!

Instead, I used the addition formula for Tan(a+b):

Tan (a+b) = (Tan (a) + Tan (b))/(1-Tan(a)Tan(b))

Using our values for Tan (a) and Tan (b) we get a numerator of 1/3 + 1/2 which equals 5/6. We get a denominator of 1-(1/3)(1/2) which also equals 5/6. So we get Tan (a+b) = 1, Tan (c) = 1 too so Tan (a+b) = Tan (c). As a,b and c are all in right angled triangles they all fall between 0 and 90 degrees, so a + b must equal c which equals 45 degrees.

A nice little mental workout. I will show some of my classes tomorrow and next week.

Here’s my back of an envelope workings:

image

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  1. June 5, 2014 at 11:02 pm

    Of course, since the same problem appears on our blog: http://fivetriangles.blogspot.com/2014/04/155-three-angle-sum.html, the implication is that it can be solved far more simply, and without trigonometry. It’s a classic problem; variants have appeared in primary school mathematics books.

    • June 6, 2014 at 6:03 am

      It can be solved very simply without trig. (ie But that’s not as fun!

      • kqrzq
        January 27, 2015 at 8:40 am

        Not as fun? A solution using trigonometry (even without a calculator) is a routine exercise. Can you solve it like a ten year old?

      • January 27, 2015 at 10:14 am

        Dunno, how would a ten year old solve it?

  2. kqrzq
    January 27, 2015 at 10:39 am

    Not at all in most cases but those who do probably wont be using trigonometry.

    Like this https://www.youtube.com/watch?v=m5evLoL0xwg ?

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