I got 3 questions to solve here.
1) if: x^2 + x + 1 = 0 then calculate: x^3333 + x^333+ x^33 + x^3 + 3 = ?
2) a^2 + b^2 + c^2 + d^2 = ab + bc + cd + da
Prove: a=b=c=d
3) You are given a circles diameter and a point that lies above the diameter. without using a protractor how would one find the EXACT perpendicular line (EXACTLY 90 degrees) from the point to the diameter.
Thanksies
I got a problem with this question. I think in projective geometry it is shown that any construction with a straightedge and compass can be done with the compass alone. But they are certainly talking about something else there. For example, you cannot draw a line with a compass! That is what you are asking, no?
Let me butt in.
Can you use a ruler or any straigth edge and a measuring device for distances if the straight edge has none?
If yes, then draw two equal line segments from the point to the diameter. Each of these equally long line segments (rays) must be longer than the shortest distance from the point to the diameter. [You can approximate that, I'm sure.]. Then get the midpoint of the line segment on the diameter that is cut by these two rays using the measuring device. Then connect this midpoint to the given point. There, you have a 90-degree angle.
[The perpendicular bisector of the base in an isosceles triangle passes through the apex of the triangle.]
Hello, Freaky-Person!
I solved #1 in a manner similar to ThePerfectHacker.
1) If , then calculate:
We are given: .
Multiply both sides by
. . and we have: .
Hence, = the three cube roots of
. . They happen to be: .
But we really don't care about their exact values,
. . we only care that: .
So we have: .
. . . . . .
You can use neither a protractor nor a compass? (You didn't mention you couldn't use a compass in your problem statement.) I presume you can't use a ruler, either, but I'm hoping you are at least allowed a straight-edge.
I was reviewing The Elements. I don't think this can be done without a compass and a straight-edge.
-Dan
As I said any construction that is doable with a straightedge and compass can be done with a compass along. Thus, if you can do it with both then you can do it with a single one.
You know there is a way to prove impossibility.I was reviewing The Elements. I don't think this can be done without a compass and a straight-edge.
you can use a straight edge, but the thing is, my teacher said that not all edges we are given are exactly 90 degrees and he said we're using bad Canadian rulers with all the edges rounded XD