Find AB and BC given AC = 62 and equal segments AB and BC.

We are given a line segment $ AC $ with point $ B $ lying between $ A $ and $ C $. The segments $ AB $ and $ BC $ are both labeled as $ 3x - 4 $. We're told that the total length $ AC = 62 $, and we are to find:

1. The value of $ x $
2. The lengths of $ AB $ and $ BC $

---

Step 1: Understand the diagram

From the diagram:

- $ AB = 3x - 4 $
- $ BC = 3x - 4 $
- So, $ AC = AB + BC $

Since $ B $ is between $ A $ and $ C $, we can write:

$$
AC = AB + BC
$$

Substitute the expressions:

$$
62 = (3x - 4) + (3x - 4)
$$

---

Step 2: Solve for $ x $

$$
62 = 3x - 4 + 3x - 4
$$
$$
62 = 6x - 8
$$

Add 8 to both sides:

$$
62 + 8 = 6x
$$
$$
70 = 6x
$$

Divide by 6:

$$
x = \frac{70}{6} = \frac{35}{3} \approx 11.67
$$

Wait — let's double-check this. Is this correct?

But hold on — let’s verify if this makes sense.

Let’s plug $ x = \frac{35}{3} $ back into $ 3x - 4 $:

$$
3x - 4 = 3 \cdot \frac{35}{3} - 4 = 35 - 4 = 31
$$

Then $ AB = 31 $, $ BC = 31 $, so $ AC = 31 + 31 = 62 $ ✔

So it checks out.

But wait — $ x = \frac{35}{3} $ is approximately 11.67, but the problem may expect an integer? Let’s check our algebra again.

Wait — did we make a mistake?

Wait:
$$
62 = 6x - 8
\Rightarrow 6x = 70
\Rightarrow x = \frac{70}{6} = \frac{35}{3}
$$

Yes, that's correct.

But maybe there’s a typo in the problem or diagram?

Wait — perhaps the labels are meant to be equal, but maybe we’re overcomplicating.

Alternatively, could the expression be different? But based on what’s shown:

- $ AB = 3x - 4 $
- $ BC = 3x - 4 $
- So $ AC = 2(3x - 4) = 6x - 8 $

Set equal to 62:

$$
6x - 8 = 62
\Rightarrow 6x = 70
\Rightarrow x = \frac{35}{3}
$$

Now compute $ AB $ and $ BC $:

$$
AB = 3x - 4 = 3 \cdot \frac{35}{3} - 4 = 35 - 4 = 31
$$

Similarly, $ BC = 31 $

So even though $ x $ is a fraction, the segment lengths are integers.

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Final Answers:

- $ x = \frac{35}{3} $
- $ AB = 31 $
- $ BC = 31 $

---

But let’s ask: Is this expected? Maybe the problem intended $ AB = 3x - 4 $, $ BC = 3x - 4 $, which implies symmetry, so $ B $ is the midpoint.

But unless the problem specifies $ x $ must be integer, this is acceptable.

Alternatively, perhaps the diagram has a typo, and the expressions are meant to be different?

But based on what’s given, this is correct.

---

✔ Final Answer:

- $ x = \frac{35}{3} $
- $ AB = 31 $
- $ BC = 31 $

---

But let’s recheck:
$ 3x - 4 = 31 $ → $ 3x = 35 $ → $ x = \frac{35}{3} $, yes.

So the answer is consistent.

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Answer:

- $ x = \boxed{\frac{35}{3}} $
- $ AB = \boxed{31} $
- $ BC = \boxed{31} $

Let me know if you'd like it written as a mixed number or decimal.