Segment Addition Postulate worksheet with examples and problems for calculating segment lengths and solving equations.

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🔷 Segment Addition Postulate:

> If point $ B $ is between points $ A $ and $ C $ on a line segment, then:
> $$
> AB + BC = AC
> $$

This means that if $ B $ lies between $ A $ and $ C $, the total length of segment $ AC $ is the sum of $ AB $ and $ BC $.

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✔ Problem 1:

> If $ DF = 10 $ and $ DE = 7 $, what is the length of $ EF $?

We assume that $ E $ is between $ D $ and $ F $, so by the Segment Addition Postulate:
$$
DE + EF = DF
$$
Substitute known values:
$$
7 + EF = 10
\Rightarrow EF = 10 - 7 = 3
$$

✔ Answer: $ EF = 3 $

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✔ Problem 2:

> If $ PO = 32 $, find the value of $ x $.
> Diagram shows: $ P \quad x \quad Q \quad 3x + 7 \quad O $

Assuming $ Q $ is between $ P $ and $ O $, then:
$$
PQ + QO = PO
\Rightarrow x + (3x + 7) = 32
$$
Simplify:
$$
4x + 7 = 32
\Rightarrow 4x = 25
\Rightarrow x = \frac{25}{4} = 6.25
$$

✔ Answer: $ x = 6.25 $

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✔ Problem 3:

> $ RA = 94 $, find $ TS $

From the diagram: $ R \quad S \quad T \quad A $
So $ RS + ST + TA = RA $, but we need more info.

Wait — likely, the labels are:
- $ RS = ? $
- $ ST = ? $
- $ TA = ? $
But only $ RA = 94 $ is given.

Wait — perhaps it's:
$ RS = 3x $, $ ST = x $, $ TA = 2x $? But this isn’t specified.

Looking at your text: “RA = 94, find TS” — and the diagram has points $ R, S, T, A $, with segments labeled.

Assuming:
- $ RS = 3x $
- $ ST = x $
- $ TA = 2x $

Then:
$$
RS + ST + TA = RA
\Rightarrow 3x + x + 2x = 94
\Rightarrow 6x = 94
\Rightarrow x = \frac{94}{6} = \frac{47}{3} \approx 15.67
$$

Then $ TS = ST = x = \frac{47}{3} $

But wait — maybe $ TS $ is $ ST $? Yes, same segment.

But unless the labels are clear, this might be incorrect.

Alternatively, perhaps it's just two segments:
- $ RS = 3x $
- $ SA = 2x $
- But $ T $ is not defined?

Wait — let’s re-read: "RA = 94, find TS"

Maybe it's: $ R \to S \to T \to A $, and the lengths are:
- $ RS = 3x $
- $ ST = x $
- $ TA = 2x $

Then total:
$$
3x + x + 2x = 6x = 94 \Rightarrow x = \frac{94}{6} = \frac{47}{3}
\Rightarrow TS = x = \frac{47}{3}
$$

✔ Answer: $ TS = \frac{47}{3} $ or $ 15.\overline{6} $

But without exact labels, this is speculative.

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✔ Problem 4:

> Solve for $ x $. Then find the length of each segment.
> Diagram: $ EC = 5x $, $ CB = 3x $, and $ EB = 56 $

Since $ C $ is between $ E $ and $ B $:
$$
EC + CB = EB
\Rightarrow 5x + 3x = 56
\Rightarrow 8x = 56
\Rightarrow x = 7
$$

Now find each segment:
- $ EC = 5x = 5(7) = 35 $
- $ CB = 3x = 3(7) = 21 $
- $ EB = 56 $ (given)

✔ Answer: $ x = 7 $, $ EC = 35 $, $ CB = 21 $

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✔ Problem 5: Complete the proof

Given:
- $ PQ = 24 $
- $ PR = 19 $
- Find $ QR $

Diagram: $ P \quad Q \quad R $, so $ Q $ is between $ P $ and $ R $

Then:
$$
PQ + QR = PR
\Rightarrow 24 + QR = 19
\Rightarrow QR = 19 - 24 = -5
$$

Wait! That can’t be. Lengths can’t be negative.

So something’s wrong.

Possibility: Maybe $ R $ is between $ P $ and $ Q $? But $ PQ = 24 $, $ PR = 19 $

If $ R $ is between $ P $ and $ Q $, then:
$$
PR + RQ = PQ
\Rightarrow 19 + RQ = 24
\Rightarrow RQ = 5
\Rightarrow QR = 5
$$

That makes sense.

So the order must be: $ P \quad R \quad Q $

Then:
- $ PR = 19 $
- $ RQ = ? $
- $ PQ = 24 $

So:
$$
PR + RQ = PQ
\Rightarrow 19 + RQ = 24
\Rightarrow RQ = 5
\Rightarrow QR = 5
$$

✔ Answer: $ QR = 5 $

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📝 Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ EF = 3 $ |
| 2 | $ x = 6.25 $ |
| 3 | $ TS = \frac{47}{3} \approx 15.67 $ (assuming $ RS=3x, ST=x, TA=2x $) |
| 4 | $ x = 7 $, $ EC = 35 $, $ CB = 21 $ |
| 5 | $ QR = 5 $ |

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✔ Explanation of Segment Addition Postulate:

The Segment Addition Postulate states that if point $ B $ lies on segment $ AC $, then:
$$
AB + BC = AC
$$
This is used to find missing lengths when a point divides a segment into two parts.

To use it:
1. Identify the three collinear points.
2. Determine which point is in the middle.
3. Add the two smaller segments to get the whole.
4. Solve algebraically if variables are involved.

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Let me know if you want me to adjust any assumptions (e.g., labels in problem 3), or if you can provide more details about the diagram!