Question

fill in the blank with the correct segment to make the statement true
ac + cf =
+ fs = bs
ab + = as
bs - bc =

Explanation:

Answer:

\( AC + CF = AF \)
\( BF + FS = BS \)
\( AB + BS = AS \) (Wait, no, let's re-examine the points. The points are A, B, C, F, S in order. So AB + BS? Wait, no, AS is the entire segment from A to S. So AB + B S? Wait, no, the segments between A-B, B-C, C-F, F-S. So AB + B C + C F + F S = AS. Wait, the third equation: AB + [ ] = AS. So AS - AB = B S? Wait, no, AS is A to S, AB is A to B, so the remaining part is B to S, so AB + BS = AS? Wait, but looking at the points: A---B---C---F---S. So from A to B is AB, then from B to S is B to C to F to S, which is BS. So AB + BS = AS? Wait, maybe I made a mistake. Wait, the third blank: AB + [ ] = AS. So the segment from B to S is BS, so AB + BS = AS? Wait, but let's check the first one: AC is A to C, CF is C to F, so AC + CF = A to F, which is AF. Correct. Second: [ ] + FS = BS. BS is B to S, FS is F to S, so the missing part is B to F, which is BF. So BF + FS = BS. Third: AB + [ ] = AS. AS is A to S, AB is A to B, so the missing part is B to S, which is BS? Wait, no, A to B to S: AB + BS = AS? Wait, but B to S is B---C---F---S, so yes. Fourth: BS - BC = [ ]. BS is B to S, BC is B to C, so subtracting BC from BS gives C to S, which is CS. Wait, let's re-express:

1. \( AC + CF = AF \) (since A to C to F is A to F)
2. \( BF + FS = BS \) (B to F to S is B to S)
3. \( AB + BS = AS \)? Wait, no, A to B to S: AB is A-B, then B to S is B-C-F-S, so AB + BS = AS? Wait, but AS is A to S, so yes. Wait, but maybe the third one is AB + B S? Wait, no, the points are A, B, C, F, S. So AB is A-B, then from B to S is B-C-F-S, so AB + BS = AS. Alternatively, maybe AB + B C + C F + F S = AS, but the blank is a single segment? Wait, no, the problem says "fill in the blank with the correct segment", so a single segment. Wait, maybe I misread. Let's list the segments:

Points in order: A, B, C, F, S.

So segments:

AB: A-B

BC: B-C

CF: C-F

FS: F-S

AC: A-C (AB + BC)

AF: A-F (AC + CF = AB + BC + CF)

AS: A-S (AB + BC + CF + FS)

BS: B-S (BC + CF + FS)

CS: C-S (CF + FS)

BF: B-F (BC + CF)

So let's re-express each equation:

1. \( AC + CF = \)? AC is A-C, CF is C-F, so A-C + C-F = A-F (AF). Correct.

1. \( [ ] + FS = BS \). BS is B-S, FS is F-S, so B-S - F-S = B-F (BF). So BF + FS = BS. Correct.

1. \( AB + [ ] = AS \). AS is A-S, AB is A-B, so A-S - A-B = B-S (BS). So AB + BS = AS. Correct.

1. \( BS - BC = \)? BS is B-S, BC is B-C, so B-S - B-C = C-S (CS). So BS - BC = CS.

So the answers are:

1. \( AC + CF = AF \)

1. \( BF + FS = BS \)

1. \( AB + BS = AS \)

1. \( BS - BC = CS \)

Wait, but let's check the third one again. AS is A to S, AB is A to B, so the segment from B to S is BS, so AB + BS = AS. Yes.

Fourth: BS is B to S, BC is B to C, so subtracting BC (B to C) from BS (B to S) gives C to S (CS). Correct.

So the blanks are:

1. AF

1. BF

1. BS

1. CS

Wait, but maybe the third one is AB + B S? Wait, no, the problem says "segment", so BS is a segment. Yes.

So the answers are:

1. AF

1. BF

1. BS

1. CS

But let's confirm with the number line:

A---B---C---F---S

So:

1. AC (A-C) + CF (C-F) = A-F (AF). Correct.

1. BF (B-F) + FS (F-S) = B-S (BS). Correct.

1. AB (A-B) + BS (B-S) = A-S (AS). Correct.

1. BS (B-S) - BC (B-C) = C-S (CS). Correct.

Yes, that makes sense.