Which of these is a correct step in constructing congruent line segments?

Question

Use a straightedge to draw two equal arcs from the endpoints.

 Use a compass to join the endpoints of the line segment.

 Use a straightedge to measure the length of the line segment. 

Use a compass to create the distance between the endpoints.

Answer 1

Hi Diamond H.,

I'll assume that you are still waiting for an answer to this, and are checking periodically for same?

So -- let's start by eliminating impossibilities:

1) You don't use a straightedge to draw arcs! Only a compass, so eliminate answer 1;

2) you can't use a compass to join endpoints, a compass only draws arcs, so out goes #2;

3) Straightedges are not rulers! Although rulers have straight edges, straightedges are only used to ensure drawing straight lines; so out goes #3;

4) So #4 is correct. You use a compass to (first set to the length of the segment #1) scribe an arc from a point of your (or as given!) choice, which will allow selection of any point on that arc to be the second endpoint. Other steps needed are: the initial gauging (tough to remember the spelling on that!) of the segment to be matched, and using a straightedge to draw the desired congruent segment from the initial endpoint you selected or were given, to cross the arc drawn and establish thereby the segment needed. Remember that congruent, for segments, means just their length, not their orientation. Just as it does for triangles, although triangles have also the ability to be "flipped" so as to establish congruence.

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Now for some mental stretching! You've seen that segments (with 1D character) may need to be translated and/or rotated to "coincide" to check congruence (or, you can measure their length!); but triangles (with 2D character) may need to be translated, rotated, or flipped to coincide to check congruence. So you've added one operation to the things you may do with a triangle (namely, flipping) for that. When you move to solid objects (in 3D), can you figure out, might you need yet another operation to do the same, namely make 2 solids coincide in space? And if so, what might that operation be? You are now entering the topic of symmetry operations; and the "operations" are now mathematical concepts which together form groups. Each appearance of the object you are manipulating is just a state; the important thing is how the operations may be strung together in sequence -- multiplied, in essence -- to show the symmetry properties of the starting object. You may get into this more in geometry, but you definitely will in chemistry, if you eventually take that as a course. It's a fascinating field.

-- Cheers, --Mr. d.