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1/ Draw a line the length you want
to get the square root of. (e.g. 7 cm) 2/ Draw an extra bit on the end that is 1 unit
long (e.g. add an extra 1 cm)
3/ Draw a semi circle that sits on your extended line. 4/ Draw a perpendicular going up to touch the semi circle.
The length of this perpendicular is your square root!
HOW DOES IT WORK?
How to CONSTRUCT a Square Root!
1/ Draw a line the length you want
to get the square root of. (e.g. 7 cm) 2/ Draw an extra bit on the end that is 1 unit
long (e.g. add an extra 1 cm)
3/ Draw a semi circle that sits on your extended line. 4/ Draw a perpendicular going up to touch the semi circle.
The length of this perpendicular is your square root!
HOW DOES IT WORK?
The big triangle in this diagram has to be right-angled
because it's inside a semi circle.
This gives us two ways of showing that this method works.
Using Pythagoras we can see that:
(x + 1)2 = c2 + a2
We can also see that:
c2 = b2 + x2
a2 = b2 + 12 = b2 + 1
Substitute the values for c2 and a2 into the first equation:
(x + 1)2 = b2 + x2 + b2 + 1
x2 + 2x + 1 = 2b2 + x2 + 1
Subtract x2 + 1 from both sides to leave:
2x = 2b2
x = b2
And therefore... √x = b
If you prefer ratios:
The big triangle and the two smaller triangles are all similar to each other.
Therefore in the two smaller triangles, the ratios of the shorter sides must be the same.
x/b = b/1
Multiply both sides by b
x = b2
And therefore...√x = b
The Prime Numbers Page
Murderous Maths Home Page
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