Question 891563: rue or False: Given a line segment, you can construct a square whose sides are congruent to the given segment.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! "Given a line segment, you can construct a square whose sides are congruent to the given segment" is true.
With a straight edge, we can
draw straight lines.
With a straight edge and a compass, we can
draw perpendicular lines, and
mark congruent segments on any line.
That is all we need to construct a square whose sides are congruent to the given segment.
First, you would draw a line, and mark a point P, in the middle of that line.
With the compass, you would measure the length of the segment, and mark that length on the line to both sides of the middle point.
 The two arcs made with the compass mark segments (AP and PB) congruent to the given segment, and the segment between the two arcs (AB) is twice as long as the given segment.
Then, with the compass, you would construct a perpendicular line through the center point (the perpendicular bisector of AB).
To draw the perpendicular bisector of AB, you would draw arcs centered at A and B, to both sides of AB, with the same opening of the compass. You would draw those arcs so that they intersect at C and D, and then you would connect C and D with a straight line.
Then you would use the compass to mark point Q on the new line, at a distance from P equal to the original segment.
Now you have P, B and Q.
PB and PQ are two sides of the square.
QB is a diagonal of the square, and triangle QBP is half of the square.
The easiest way to complete the square is to use the compass to mark point R,
to the other side of QB, and at a distance from B and Q equal to the length of the original segment. For that, you would use the compass to mark intersecting arcs centered at Q and B, with the length of the original segment for a radius:
Now you have P, B, Q and R, the four vertices of the square.
All that is left to do is connect R to Q and to B:
|
|
|
