Today's question is not from Yahoo Answers, but rather from my head. Well, not exactly from my head. I read about the possibility to construct a square root of a number, but I didn't find any instructions about how to do it. So, here is what I found out.
There is a line of length x. Construct a line of length sqrt(x)
If you remember anything about averages, there are two common types. There is a simple arithmetic average, simply adding two numbers and halving the result. There is also a geometric average, which is multiplying two numbers and taking the square root of the result.
This average is called a geometric average because it happens in, surprisingly, geometry. In a right triangle, the altitude from the 90 degree angle to the hypotenuse is equal to the square root of the two parts of the hypotenuse created from the altitude. If the altitude creates segments of length x and y, then the altitude's length is sqrt(xy). We can use that to find sqrt(x). All we have to do is make y = 1.
So to construct this segment of length sqrt(x) we construct a segment with length x and a segment with length 1 (the 1 is relative to x. If x is 5, the segment of length 1 must be 5 times shorter). Now we created the hypotenuse. Now construct the altitude to the hypotenuse from the point the two segments are joined.
We are in a little trouble now (a trouble that made me think for almost 30 minutes). We need to find the exact point on the altitude that will create a right angle with the endpoints of the hypotenuse. However, we have no way of knowing where is that point, since we construct a square root. Unless it's a square root of a perfect square, we cannot count units of 1 and create the right triangle from there.
After almost 30 minutes of thinking, it hit me. When a right triangle is inscribed inside a circle, the hypotenuse is a diameter. Now we can create an exact right triangle and an exact square root. Find the midpoint of the hypotenuse and create a circle around it.
Now it's easy. Continue the altitude until it intersects the circle. Then you can complete the triangle and take the altitude. Square root found.
Hope you liked it. Now go test it - find the square roots of numbers like 4 and 9 and see if you get 2 and 3. If you do, you construct well.
Yours,
Nadav
nadavs
Saturday, May 31, 2008
Constructing a Square Root
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5 comments:
how do you make y = 1?
i.e.
are you sure you can always construct a line that's x times shorter than x?
After the paragraph that mentions y = 1, there is an explanation: for example, if we want the square root of 5, we need to make one segment five times longer than the other.
You can't always construct a line which is x times shorter than x. If x is an irrational number, like pi, there is a high chance you won't be able to do so (there are a few exceptions, though).
nadavs
Good fill someone in on and this fill someone in on helped me alot in my college assignement. Say thank you you as your information.
Nice post...
How about constructing a length of x^2? Will you help me.
how do you construct a length of root 7
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