Current Problem of the Week
June 17-21, 1996
Clint Joseph with Ryan Cayford
The problem I am currently faced with is "How long is a diagonal line from each of the vertices of a piece of paper?" A diagram of this is represented below. The only measurements are provided in my diagram.
For me to complete this task successfully I will need to redraw the above diagram several times, look for any symmetrical lines, be able to notice number patterns, divide the page into shapes and know Pythagoras' rule.
I first redrew the question and added some of my own letters to find the correct answers. My new diagram is below.
I then proceeded to provide the class with some simple mathematical facts about the above diagram. They were as follows:
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? = 18 squared + a squared
b = 18 squared + a squared
b squared - a squared = 18 squared
a + b = 24
From here I continued on my way to investigate further into this problem. I proceeded to find the value of "a". I already knew a + b = 24. I was extremely lucky in this situation and after only two attempts I found that a = 5.25 and b = 18.75.
I then squared the two numbers a = 27.5625 and b = 351.5625 and subtracted these two totals and the answer fortunately equalled 18 squared. I then had to find the subtraction answer to 18.75 - 5.25 = 13.50.
To find the exact answer one small step was still to be taken. I had to know 18 + 13.5 = 22.5 and that was the full length of the diagonal line. Pythagoras' rule came into process here, being h squared = a squared + b squared.
The answer was much easier to obtain after I split the page into a triangle like the one below.
I then checked my answer by drawing the question exactly to size and my accuracy was absolutely brilliant (Spot On).
This fully proves that the answer of the question above is 22.5cm.
South Fremantle Senior High School