Geometry problem 15: Find CE using the segment addition postulate with given expressions 3x + 47, 10, 27 + x, and x + 26.

Let's solve this step-by-step using the Segment Addition Postulate, which states that if points $ A $, $ B $, and $ C $ are collinear and $ B $ is between $ A $ and $ C $, then:

$$
AB + BC = AC
$$

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Given:

We have a line segment $ BE $ with points in order:
$$
B \quad — \quad C \quad — \quad D \quad — \quad E
$$

We're given:

- $ BC = 3x + 47 $
- $ CD = ? $ (not directly given)
- $ DE = 10 $
- $ BD = 27 + x $
- $ CE = ? $ (this is what we need to find)

Also, there’s a note showing:
- $ BD + DE = BE $
- $ BC + CE = BE $

And another expression: $ x + 26 $ is labeled from point $ D $ to $ E $? Wait — let's clarify.

Wait! The label $ x + 26 $ is shown as the length from $ D $ to $ E $? But we're told $ DE = 10 $. That seems conflicting.

Let’s carefully analyze the diagram:

- From $ B $ to $ C $: $ BC = 3x + 47 $
- From $ B $ to $ D $: $ BD = 27 + x $
- From $ D $ to $ E $: $ DE = 10 $
- So, $ BE = BD + DE = (27 + x) + 10 = 37 + x $

But also, $ BE = BC + CE $

So:
$$
BC + CE = BE
\Rightarrow (3x + 47) + CE = 37 + x
$$

Wait — that would give:
$$
CE = (37 + x) - (3x + 47) = 37 + x - 3x - 47 = -2x - 10
$$

That gives a negative length, which can't be right. So something's wrong.

Wait — perhaps the labeling is off.

Let’s look again.

We have:

- $ BC = 3x + 47 $
- $ BD = 27 + x $
- $ DE = 10 $
- So $ BE = BD + DE = (27 + x) + 10 = 37 + x $

But $ BD = BC + CD $, so:
$$
BD = BC + CD
\Rightarrow 27 + x = (3x + 47) + CD
\Rightarrow CD = (27 + x) - (3x + 47) = 27 + x - 3x - 47 = -2x - 20
$$

Again, negative? That can't be.

So clearly, either the expressions are mislabeled or our assumption about the order is wrong.

Wait — maybe $ BC = 3x + 47 $ is too long? But $ BD = 27 + x $, which is smaller than $ BC $ if $ x $ is small?

But $ C $ is between $ B $ and $ D $, so $ BC < BD $, but here $ BC = 3x + 47 $, $ BD = 27 + x $. So unless $ 3x + 47 < 27 + x $, that would mean:

$$
3x + 47 < 27 + x \Rightarrow 2x < -20 \Rightarrow x < -10
$$

Which would make lengths negative — not possible.

So contradiction.

Ah! Wait — maybe the label $ 3x + 47 $ is not $ BC $? Let's check the diagram again.

Looking at the image:

- Above the segment $ BC $, it says $ 3x + 47 $
- Below, from $ B $ to $ D $, it says $ 27 + x $
- From $ D $ to $ E $, it says $ 10 $
- And from $ D $ to $ E $, there’s a yellow circle around $ x + 26 $

Wait — that’s critical.

There’s a blue arrow from $ D $ to $ E $ labeled $ x + 26 $, and also written above it as $ 10 $.

So likely, $ DE = 10 $, but also $ DE = x + 26 $? That must be the key.

So we can set up an equation:

$$
DE = x + 26 = 10
\Rightarrow x = 10 - 26 = -16
$$

But $ x = -16 $? Then plug into other expressions:

- $ BC = 3x + 47 = 3(-16) + 47 = -48 + 47 = -1 $ → negative length!

Not possible.

So contradiction again.

Wait — maybe the $ x + 26 $ is not $ DE $? Look again.

The yellow circle is around $ x + 26 $, and it's under the segment from $ D $ to $ E $, but the arrow goes from $ D $ to $ E $, and it says $ x + 26 $. But above it, it says $ 10 $. So likely, $ DE = x + 26 $, and $ DE = 10 $, so:

$$
x + 26 = 10 \Rightarrow x = -16
$$

But again, leads to negative lengths.

Alternatively, maybe $ x + 26 $ is not $ DE $, but $ CE $? But no — the arrow is from $ D $ to $ E $.

Wait — perhaps the $ x + 26 $ is not $ DE $, but the total from $ C $ to $ E $? But the arrow starts at $ D $, ends at $ E $.

Wait — maybe the label $ x + 26 $ is meant to be the length from $ C $ to $ E $? But the arrow is drawn from $ D $ to $ E $, so it should be $ DE $.

Unless the diagram has a typo.

Let’s re-express what we know.

From the image:

- Points: $ B $ — $ C $ — $ D $ — $ E $
- $ BC = 3x + 47 $
- $ BD = 27 + x $
- $ DE = 10 $
- Also, there's a label $ x + 26 $ under the segment $ DE $, suggesting $ DE = x + 26 $

So we must conclude that:

$$
DE = x + 26 = 10 \Rightarrow x = -16
$$

Then:

- $ BC = 3(-16) + 47 = -48 + 47 = -1 $ → impossible

So contradiction.

Therefore, the only logical conclusion is that the label $ x + 26 $ is not $ DE $, but rather $ CE $, and the arrow is misdrawn?

Wait — the arrow for $ x + 26 $ starts at $ D $ and ends at $ E $, so it must be $ DE $.

But $ DE $ is also labeled as $ 10 $, so they must be equal.

So unless $ x + 26 = 10 $, but that gives $ x = -16 $, which makes $ BC = 3(-16)+47 = -1 $, invalid.

So something is wrong.

Wait — perhaps the expression $ 3x + 47 $ is not $ BC $? Let’s read the diagram carefully.

It says: from $ B $ to $ C $: $ 3x + 47 $

From $ B $ to $ D $: $ 27 + x $

Since $ C $ is between $ B $ and $ D $, then:

$$
BD = BC + CD
\Rightarrow 27 + x = (3x + 47) + CD
\Rightarrow CD = (27 + x) - (3x + 47) = -2x - 20
$$

But $ CD $ must be positive, so $ -2x - 20 > 0 \Rightarrow -2x > 20 \Rightarrow x < -10 $

But then $ BC = 3x + 47 $, if $ x < -10 $, say $ x = -15 $, then $ BC = -45 + 47 = 2 $, OK

$ BD = 27 + (-15) = 12 $

$ CD = BD - BC = 12 - 2 = 10 $, so $ CD = 10 $

Now, $ DE = 10 $, so $ CE = CD + DE = 10 + 10 = 20 $

But we need to find $ CE $, and we have $ x = -15 $? But earlier we had $ DE = x + 26 $, so if $ DE = 10 $, then $ x + 26 = 10 \Rightarrow x = -16 $

But if $ x = -16 $, then $ BC = 3(-16) + 47 = -48 + 47 = -1 $ → invalid

So inconsistency.

Wait — perhaps the $ x + 26 $ is not $ DE $, but $ CE $? But the arrow is from $ D $ to $ E $, so it can't be $ CE $.

Unless the label $ x + 26 $ is meant to be $ CE $, and the arrow is just poorly placed.

Let’s suppose that $ CE = x + 26 $, and $ DE = 10 $, and $ CD = ? $

We know:

- $ BC = 3x + 47 $
- $ BD = 27 + x $
- $ BD = BC + CD \Rightarrow CD = BD - BC = (27 + x) - (3x + 47) = -2x - 20 $
- $ CE = CD + DE = (-2x - 20) + 10 = -2x - 10 $
- But also $ CE = x + 26 $ (from the yellow circle)

So set them equal:

$$
-2x - 10 = x + 26
\Rightarrow -2x - x = 26 + 10
\Rightarrow -3x = 36
\Rightarrow x = -12
$$

Now test values:

- $ BC = 3(-12) + 47 = -36 + 47 = 11 $
- $ BD = 27 + (-12) = 15 $
- $ CD = BD - BC = 15 - 11 = 4 $
- $ DE = 10 $
- $ CE = CD + DE = 4 + 10 = 14 $
- Also $ CE = x + 26 = -12 + 26 = 14 $ → matches!

Perfect!

So even though $ x = -12 $, all lengths are positive.

Now, the problem asks: Find $ CE $

We found $ CE = 14 $

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✔ Final Answer:

$$
\boxed{14}
$$

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Explanation:

We used the Segment Addition Postulate:

- $ BD = BC + CD $
- $ CE = CD + DE $

Given:
- $ BC = 3x + 47 $
- $ BD = 27 + x $
- $ DE = 10 $
- $ CE = x + 26 $ (from the label, even though it's on the $ DE $ segment, it's likely meant to be $ CE $, or the diagram is mislabeled — but mathematically, it works if $ CE = x + 26 $)

So:
1. $ CD = BD - BC = (27 + x) - (3x + 47) = -2x - 20 $
2. $ CE = CD + DE = (-2x - 20) + 10 = -2x - 10 $
3. Set equal to $ x + 26 $:
$$
-2x - 10 = x + 26 \Rightarrow -3x = 36 \Rightarrow x = -12
$$
4. Then $ CE = x + 26 = -12 + 26 = \boxed{14} $

✔ All lengths are positive and consistent:
- $ BC = 11 $
- $ CD = 4 $
- $ DE = 10 $
- $ BD = 15 $
- $ BE = BD + DE = 15 + 10 = 25 $
- $ CE = CD + DE = 4 + 10 = 14 $

All good.

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Final Answer: $ \boxed{14} $