Centroid Worksheet - Free Printable

Let’s solve each problem step by step. We’re working with centroids of triangles — the point where all three medians intersect. A key fact:
> The centroid divides each median into two parts, with the part from the vertex to the centroid being twice as long as the part from the centroid to the midpoint of the opposite side.

So if a median is split by the centroid G, then:
- From vertex to G = 2 × (from G to midpoint)
- Total median length = 3 × (from G to midpoint)

We’ll use this for every problem.

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Triangle JKL: Medians are JE, KF, LD → they meet at G

#### Problem 1: If GF = 15, then KG = ?

GF is from centroid G to midpoint F on JL. So KG (from vertex K to G) should be twice that.

→ KG = 2 × GF = 2 × 15 = 30

✔ Check: Median KF = KG + GF = 30 + 15 = 45 → makes sense.

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#### Problem 2: If JG = 13, then JE = ?

JG is from vertex J to centroid G. Since centroid splits median in 2:1 ratio (vertex to centroid : centroid to midpoint), then:

JE = JG + GE
But GE = half of JG? No — wait: JG is the longer part (2 parts), so GE is 1 part.

So if JG = 13 → that’s 2 parts → 1 part = 6.5 → GE = 6.5

Then JE = JG + GE = 13 + 6.5 = 19.5

Alternatively: total median = 3 parts → JG = 2 parts → so whole median = (3/2) × JG = (3/2)×13 = 19.5

✔ Answer: 19.5

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#### Problem 3: If JL = 22, then FL = ?

F is the midpoint of JL (since KF is a median). So FL is half of JL.

FL = JL ÷ 2 = 22 ÷ 2 = 11

✔ Simple!

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#### Problem 4: If KE = 20, then KL = ?

E is the midpoint of KL (because JE is a median from J to side KL).

So KE = EL = half of KL → KL = 2 × KE = 2 × 20 = 40

✔ Done.

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#### Problem 5: If DL = 24, LG = ? and DG = ?

DL is the full median from D to L? Wait — let’s check triangle JKL.

Medians: JE, KF, LD → so LD goes from L to D, which must be midpoint of JK.

Wait — actually, in triangle JKL, median LD means D is midpoint of JK, and L is vertex.

So median is from L to D (midpoint of JK). Centroid G lies on LD.

Given DL = 24 → that’s the entire median from D to L? But usually we say “median from vertex”, so probably DL means from D to L — but D is midpoint, L is vertex → so yes, median length is 24.

Centroid G divides it such that LG (from vertex L to G) is 2/3 of the median, and GD (from G to midpoint D) is 1/3.

Wait — notation: DL = 24 → distance from D to L is 24. Since L is vertex, D is midpoint, then:

LG = (2/3) × DL = (2/3) × 24 = 16

DG = (1/3) × DL = (1/3) × 24 = 8

Check: 16 + 8 = 24 → correct.

✔ Answers: LG = 16, DG = 8

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Now Triangle DEF: Medians DN, EO, FM → meet at P

Same rule applies: centroid P divides each median 2:1 (vertex to P : P to midpoint)

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#### Problem 6: If DE = 45, then DM = ?

M is midpoint of EF? Wait — median FM → so M is midpoint of DE? Let’s see.

In triangle DEF, medians are DN, EO, FM.

- DN: from D to N → N must be midpoint of EF
- EO: from E to O → O must be midpoint of DF
- FM: from F to M → M must be midpoint of DE

Yes! So M is midpoint of DE.

DE = 45 → DM = half of DE = 45 ÷ 2 = 22.5

✔ Answer: 22.5

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#### Problem 7: If PD = 20, then PN = ?

PD is from centroid P to vertex D? Or to midpoint?

Median is DN → from D (vertex) to N (midpoint of EF). Centroid P is on DN.

If PD = 20 — is that from P to D or D to P? Probably same thing.

Assuming PD means distance from P to D → since D is vertex, then PD is the longer part (2 parts).

Then PN (from P to N, midpoint) is 1 part → so PN = PD ÷ 2 = 20 ÷ 2 = 10

✔ Answer: 10

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#### Problem 8: If DO = 8.5, then FD = ?

O is midpoint of DF (since EO is median from E to O on DF).

DO = 8.5 → since O is midpoint, then FD = 2 × DO = 2 × 8.5 = 17

✔ Answer: 17

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#### Problem 9: If OE = 22, EP = ? and OP = ?

OE is the full median from O to E? Wait — median is EO → from E (vertex) to O (midpoint of DF).

So OE = 22 → that’s the whole median.

Centroid P divides it: EP (from vertex E to P) = 2/3 of OE
OP (from P to midpoint O) = 1/3 of OE

EP = (2/3) × 22 = 44/3 ≈ 14.666... → better as fraction: 44/3

OP = (1/3) × 22 = 22/3

But maybe they want decimals? Let’s keep fractions unless specified.

Actually, 22 ÷ 3 = 7.333..., so:

EP = 14.666... → but perhaps write as mixed number? Or decimal?

Since original numbers are integers or halves, maybe leave as fractions.

But let’s check: 22 is divisible by... no, so:

EP = 44/3, OP = 22/3

But maybe simplify? They’re already simplified.

Alternatively, write as decimals rounded? But problem doesn’t specify.

Looking back at other problems — problem 2 had 19.5, so decimals are okay.

So:

EP = (2/3)*22 = 44/3 ≈ 14.67 (if rounding to hundredths)

But exact value is better.

Wait — perhaps I misread: "If OE = 22" — is OE the whole median? Yes.

But in notation, sometimes OE might mean from O to E — same as EO.

So yes.

To avoid confusion, let's compute exactly:

EP = 2/3 * 22 = 44/3 = 14 2/3

OP = 1/3 * 22 = 22/3 = 7 1/3

But since the worksheet has blanks, probably expect decimal or fraction.

Looking at problem 5, they used integers. Problem 2 had 19.5.

Perhaps write as decimals: 14.67 and 7.33? But that’s approximate.

Better to use fractions or exact decimals.

44/3 = 14.666... repeating — not nice.

Wait — did I make a mistake?

Another thought: maybe "OE = 22" means from O to E, but O is midpoint, E is vertex — so same as before.

Perhaps the problem intends OE to be the segment from O to E, which is the whole median.

I think we have to go with fractions.

But let me see problem 10 — MP=10, find PF — similar.

Perhaps they expect fractional answers.

But to match style, let's use decimals with one place if needed.

44/3 = 14.666... → round to 14.7? Not good.

Wait — perhaps I can leave as improper fractions.

But looking at the worksheet, blank lines suggest simple numbers.

Double-check: is OE the whole median? Yes.

Centroid divides it 2:1, so EP:PO = 2:1

So EP = 2k, PO = k, total 3k = 22 → k = 22/3

So EP = 44/3, OP = 22/3

I think we should write them as fractions.

But let's proceed and see.

For now, I'll write:

EP = \frac{44}{3}, OP = \frac{22}{3}

But perhaps simplify? No.

Or write as mixed numbers: 14 2/3 and 7 1/3

That might be acceptable.

Since problem 2 had 19.5, which is decimal, perhaps here too.

19.5 is 39/2, so fractions are fine.

I'll use fractions for accuracy.

✔ EP = 44/3, OP = 22/3

But let's confirm with next problem.

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#### Problem 10: If MP = 10, then PF = ?

Median is FM → from F (vertex) to M (midpoint of DE). Centroid P is on FM.

MP = 10 — is that from M to P or P to M? Same.

M is midpoint, F is vertex.

So if MP = 10, that’s from midpoint M to centroid P → which is the shorter part (1 part).

Then PF (from P to vertex F) is 2 parts → so PF = 2 × MP = 2 × 10 = 20

✔ Answer: 20

This confirms our understanding.

Back to problem 9: OE = 22 (whole median), so EP = 2/3 * 22 = 44/3, OP = 1/3 * 22 = 22/3

We can write as decimals if needed, but since 22 isn't divisible by 3, fractions are precise.

Perhaps the problem expects us to leave it as is.

I think for consistency, I'll use fractions.

But let's see the answer format — probably numerical values.

Another idea: maybe "OE = 22" means the length from O to E, but in some contexts, it might be different — but I think it's standard.

Perhaps calculate numerically.

I recall that in some worksheets, they accept fractions.

So I'll go with:

EP = \frac{44}{3} or approximately 14.67, but better exact.

To match the style of problem 2 (which was 19.5), perhaps write as decimal.

19.5 is exact, while 44/3 is not.

But 22/3 is about 7.333, so perhaps write 7.3 or something — no.

I think it's fine to use fractions.

Let me check online or standard practice — but since I can't, I'll assume fractions are acceptable.

For the sake of this, I'll write the answers as fractions.

So for problem 9:

EP = 44/3, OP = 22/3

But let's write them as mixed numbers for clarity: 14 2/3 and 7 1/3

That might be better.

Yes.

So EP = 14 2/3, OP = 7 1/3

Now, compiling all answers.

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Final Answers:

1. KG = 30
2. JE = 19.5
3. FL = 11
4. KL = 40
5. LG = 16, DG = 8
6. DM = 22.5
7. PN = 10
8. FD = 17
9. EP = 14 2/3, OP = 7 1/3 (or 44/3 and 22/3)
10. PF = 20

For problem 9, to be consistent with decimal usage in problem 2 and 6, perhaps convert to decimals.

Problem 2: 19.5 — which is exact.

Problem 6: 22.5 — exact.

Problem 9: 44/3 = 14.666... — not terminating, so better as fraction.

But in many school contexts, they might expect decimal approximation.

However, since the problem didn't specify, and to be precise, I'll use fractions.

But let's see the blank — it says "EP = _________ and OP = _________" — so two blanks.

Perhaps write as decimals rounded to nearest tenth.

44/3 ≈ 14.7, 22/3 ≈ 7.3

But 14.7 is 147/10, not exact.

I think it's safer to use fractions.

Upon second thought, in the context of centroids, ratios are exact, so fractions are appropriate.

So I'll keep them as fractions.

But to write neatly:

EP = \frac{44}{3}, OP = \frac{22}{3}

Or in mixed numbers: 14\frac{2}{3}, 7\frac{1}{3}

I think mixed numbers are more student-friendly.

Yes.

So final decision:

For problem 9: EP = 14 2/3, OP = 7 1/3

Now, writing all answers clearly.

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Final Answer:
1. 30
2. 19.5
3. 11
4. 40
5. LG = 16, DG = 8
6. 22.5
7. 10
8. 17
9. EP = 14 2/3, OP = 7 1/3
10. 20