Solve the problem. Given: AE = 42 cm, BC = 1 cm, CD = DE = 7 cm, EF = 8 cm, ET = 12 cm. Find: AB, FT, AT.

Solution:

We are given a line segment with points A, B, C, D, E, F, T. We need to find the lengths of segments AB, FT, and AT.

First, let's find AB:

We know that AE = 42 cm. We can express AE as the sum of its segments: AE = AB + BC + CD + DE + EF + FT.

However, it is easier to find AB by subtracting the known segments from AE. Since CD = DE = 7 cm, then CE = CD + DE = 7 + 7 = 14 cm.

We also know that ET = 12 cm and EF = 8 cm. So, FT = ET - EF = 12 - 8 = 4 cm.

Now, let's reconsider AE. We have AE = AB + BC + CD + DE + EF + FT. We can also say that AE = AC + CE + EF + FT. Or AE = AB + BE + EF + FT.

Let's use the information we have: AE = 42 cm, BC = 1 cm, CD = 7 cm, DE = 7 cm, EF = 8 cm, ET = 12 cm.

We can find CT by summing CD, DE, and EF: CT = CD + DE + EF = 7 + 7 + 8 = 22 cm.

We can find CE = CD + DE = 7 + 7 = 14 cm.

We can find CF = CE + EF = 14 + 8 = 22 cm.

We can find AT by summing all segments: AT = AB + BC + CD + DE + EF + FT.

Let's find AB first. We know AE = 42 cm. We also know that AE = AC + CE + EF + FT. This is not directly helping.

Let's try to express AE in terms of segments we can calculate or are given directly. AE = AB + BC + CE + EF + FT.

We know ET = 12 cm and EF = 8 cm. Therefore, FT = ET - EF = 12 - 8 = 4 cm.

We know DE = 7 cm and EF = 8 cm. We cannot directly find DF.

Let's find the length of the segment starting from A to E, using the given lengths.

AE = 42 cm.

Let's find the lengths of the segments from right to left.

FT = ET - EF = 12 cm - 8 cm = 4 cm.

EF = 8 cm.

DE = 7 cm.

CD = 7 cm.

BC = 1 cm.

Now we can find AB. The total length AE is the sum of all segments: AE = AB + BC + CD + DE + EF + FT.

42 = AB + 1 + 7 + 7 + 8 + 4.

42 = AB + 27.

AB = 42 - 27 = 15 cm.

Now let's find FT. We already calculated FT = ET - EF = 12 cm - 8 cm = 4 cm.

Finally, let's find AT. AT = AB + BC + CD + DE + EF + FT.

AT = 15 + 1 + 7 + 7 + 8 + 4 = 42 cm.

Wait, AT is the same as AE. Let's check the points. A, B, C, D, E, F, T. So AT = AE.

Therefore, AT = 42 cm.

Check:

AB = 15 cm

BC = 1 cm

CD = 7 cm

DE = 7 cm

EF = 8 cm

FT = 4 cm

Sum: 15 + 1 + 7 + 7 + 8 + 4 = 42 cm. This matches AE.

Also, ET = EF + FT = 8 + 4 = 12 cm, which is given.

Final calculations:

1. Find FT:

\( FT = ET - EF = 12 \text{ cm} - 8 \text{ cm} = 4 \text{ cm} \)

2. Find AB:

\( AE = AB + BC + CD + DE + EF + FT \)

\( 42 \text{ cm} = AB + 1 \text{ cm} + 7 \text{ cm} + 7 \text{ cm} + 8 \text{ cm} + 4 \text{ cm} \)

\( 42 \text{ cm} = AB + 27 \text{ cm} \)

\( AB = 42 \text{ cm} - 27 \text{ cm} = 15 \text{ cm} \)

3. Find AT:

Since the points are in order A, B, C, D, E, F, T, the segment AT is the entire segment AE.

\( AT = AE = 42 \text{ cm} \)

Answer:

AB = 15 cm, FT = 4 cm, AT = 42 cm.