Geometry Worksheets | Triangle Worksheets - Free Printable

This worksheet focuses on the properties of medians in a triangle and their point of intersection, called the centroid.

Key Concept: The Centroid

The centroid (labeled 'C' in all the diagrams) is the point where all three medians of a triangle intersect. A crucial property of the centroid is that it divides each median into two segments in a 2:1 ratio, with the longer segment being the part from the vertex to the centroid.

In other words, for any median:
- The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.
- So, if we let the entire median length be `M`, then:
- Vertex to Centroid = `(2/3) * M`
- Centroid to Midpoint = `(1/3) * M`

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Now, let's solve each problem using this 2:1 ratio property.

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Problem 1: Find YN if UN = 10.

- UN is a median from vertex U to the midpoint N of side QN.
- The centroid C divides UN into UC and CN.
- We are asked to find YN. Looking at the diagram, YN is the segment from vertex Y to the centroid C.
- Wait, let's re-examine the diagram. The median from Y goes to the midpoint of side UN, which is point V. So the full median is YV.
- The centroid C lies on YV, dividing it into YC and CV.
- The question asks for YN, but N is the midpoint of QN, not related to the median from Y. This seems like a labeling error or misprint.
- However, looking closely, the line segment labeled "YN" is actually the median from vertex Y to the midpoint of side UN (which is point V). But the label "YN" is written along the segment from Y to C. This is confusing.

Let's assume the intended question is: Find the length of the segment from vertex Y to the centroid C (YC), given that the full median UN = 10.

But that doesn't make sense because YC is part of a different median.

Alternatively, perhaps "YN" is meant to be the segment from Y to the centroid C, and UN=10 is given to find something else? That doesn't fit.

Wait — another possibility: Perhaps "YN" is a typo and should be "CN" or "UC". Let's look at the diagram again.

Actually, in the first diagram, the median from U is UN. The centroid C divides UN into UC and CN. Since UN = 10, and C divides it in a 2:1 ratio (vertex to centroid : centroid to midpoint), then:

- UC = (2/3) * 10 = 20/3 ≈ 6.67
- CN = (1/3) * 10 = 10/3 ≈ 3.33

But the question asks for "YN". Looking at the diagram, YN is drawn as the median from Y to the midpoint of UN (which is V). So YN is not the same as UN.

Perhaps "YN" is meant to be the segment from Y to C? In that case, we need the length of the median from Y, which is not given.

This is ambiguous. Let's assume that "YN" refers to the segment from the vertex Y to the centroid C, and since no other information is given, perhaps the problem intends for us to realize that without the length of the median from Y, we cannot find YC.

But that can't be right for a worksheet.

Another interpretation: Maybe "YN" is a mislabel, and it should be "CN", the segment from the centroid to the midpoint N.

Given UN = 10, and C divides UN in 2:1, with the shorter part being from centroid to midpoint, then:

CN = (1/3) * UN = (1/3) * 10 = 10/3

But the blank is labeled "YN", not "CN".

Perhaps in the diagram, the point labeled "N" is actually the midpoint of the side opposite Y, so YN is the full median from Y. But then UN=10 is irrelevant.

I think there might be a labeling issue. To proceed, I'll assume that "YN" is meant to be the segment from the centroid to the midpoint N, i.e., CN.

So, YN = CN = (1/3) * UN = 10/3

But to be precise, let's check the other problems; they seem more straightforward.

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Problem 2: Find JQ if KQ = 16.

- KQ is part of the median from vertex Q to the midpoint K of side JD.
- The centroid C divides QK into QC and CK.
- We are given KQ = 16. Note that KQ is the same as QK, so the full median from Q to K is 16.
- The centroid C divides this median in a 2:1 ratio, with QC being twice CK.
- So, QC = (2/3) * 16 = 32/3
- CK = (1/3) * 16 = 16/3

But the question asks for JQ. Looking at the diagram, JQ is the side of the triangle, not a median. That doesn't make sense.

Wait, perhaps "JQ" is a typo and should be "QC" or "CK".

Alternatively, maybe "JQ" is meant to be the segment from J to Q, but that's a side, and we have no information about sides.

Another possibility: In the diagram, the median from J goes to the midpoint of DQ, which is M. So the median is JM.

The question says "Find JQ if KQ = 16". KQ is given, which is part of the median from Q.

Perhaps "JQ" is meant to be the segment from J to the centroid C, i.e., JC.

But again, we don't have the length of the median from J.

This is problematic.

Let's look at problem 3.

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Problem 3: SP = 18. Find CS.

- SP is a median from vertex S to the midpoint P of side UY.
- The centroid C divides SP into SC and CP.
- Given SP = 18.
- The centroid divides the median in a 2:1 ratio, with the part from the vertex to the centroid being twice the part from the centroid to the midpoint.
- So, SC = (2/3) * SP = (2/3) * 18 = 12
- CP = (1/3) * SP = 6

The question asks for CS. CS is the same as SC, so CS = 12.

✔ This one is clear.

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Problem 4: CH = 17. Find JH.

- CH is given as 17. Looking at the diagram, CH is part of the median from H to the midpoint of JS, which is M. So the full median is HM.
- C is the centroid, so it divides HM into HC and CM.
- Given CH = 17. CH is the same as HC, so the segment from the vertex H to the centroid C is 17.
- Since the centroid divides the median in a 2:1 ratio, with vertex to centroid being 2 parts, then:
- HC = 2 parts = 17
- So 1 part = 17/2 = 8.5
- Therefore, the full median HM = 3 parts = 3 * 8.5 = 25.5
- And CM = 1 part = 8.5

But the question asks for JH. JH is the side from J to H, not a median. That doesn't make sense.

Perhaps "JH" is meant to be "HM" or "CM".

Another possibility: In the diagram, the median from J goes to M, so JM is the median. JH is not a median.

Perhaps "JH" is a typo and should be "JM" or "JC".

Given that CH = 17 is the vertex to centroid part, and we need to find something else, but JH is not related.

Let's assume that "JH" is meant to be the full median from J, but we have no information about it.

This is confusing.

Perhaps "JH" is meant to be "CH" itself, but that's given.

I think there might be a consistent labeling issue.

Let's look at problem 5.

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Problem 5: BY = 51. Find CY.

- BY is a median from vertex B to the midpoint Y of side PS.
- The centroid C divides BY into BC and CY.
- Given BY = 51.
- The centroid divides the median in a 2:1 ratio, with BC being twice CY.
- So, CY = (1/3) * BY = (1/3) * 51 = 17

✔ This is clear. CY = 17

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Problem 6: Find OZ if RZ = 14.

- RZ is part of the median from R to the midpoint Z of side TZ? Wait, the triangle is TRZ? Looking at the diagram, the triangle is TZR, with vertices T, Z, R.
- The median from R goes to the midpoint of TZ, which is O. So the full median is RO.
- RZ is given as 14. But Z is a vertex, not the midpoint. This doesn't make sense.

Wait, in the diagram, Z is a vertex, and O is the midpoint of TZ. The median from R is RO.

RZ is the side from R to Z, not a median.

Perhaps "RZ" is meant to be "RO", the full median.

Assume that "RZ" is a typo and should be "RO" = 14.

Then, the centroid C divides RO into RC and CO.

We are asked to find OZ. OZ is the segment from O to Z, which is half of TZ, since O is the midpoint. But we have no information about TZ.

This is not working.

Perhaps "OZ" is meant to be "CO", the segment from the centroid to the midpoint O.

If RO = 14 (assuming RZ is a typo), then CO = (1/3) * RO = 14/3.

But the question asks for OZ.

Another possibility: In the diagram, the median from Z goes to the midpoint of TR, which is Y. So the median is ZY.

The question says "Find OZ if RZ = 14". RZ is a side.

I think there's a consistent issue with labeling.

Let's look at problem 7.

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Problem 7: Find JA if TA = 12.

- TA is a median from vertex T to the midpoint A of side LJ.
- The centroid C divides TA into TC and CA.
- Given TA = 12.
- The centroid divides the median in a 2:1 ratio, with TC = (2/3)*12 = 8, and CA = (1/3)*12 = 4.

The question asks for JA. JA is the segment from J to A. A is the midpoint of LJ, so JA is half of LJ, but we have no information about LJ.

Perhaps "JA" is meant to be "CA" or "TC".

Another possibility: In the diagram, the median from J goes to the midpoint of LA, which is Q. So the median is JQ.

We are given TA=12, which is a different median.

Perhaps "JA" is meant to be the segment from J to the centroid C, i.e., JC.

But again, we don't have the length of the median from J.

This is frustrating.

Let's look at problem 8.

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Problem 8: HA = 12. Find CH.

- HA is given as 12. Looking at the diagram, HA is part of the median from H to the midpoint A of side DL.
- So the full median is HD? No, from H to A, so the median is HA.
- The centroid C divides HA into HC and CA.
- Given HA = 12.
- The centroid divides the median in a 2:1 ratio, with HC being twice CA.
- So, HC = (2/3) * HA = (2/3) * 12 = 8
- CA = (1/3) * 12 = 4

The question asks for CH, which is the same as HC, so CH = 8.

✔ This is clear.

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## Summary of Clear Problems:

From the above, problems 3, 5, and 8 are clear and can be solved directly.

For the others, there appear to be labeling issues or typos.

However, based on the pattern and the fact that in most cases, the given length is the full median or the vertex-to-centroid part, and the question asks for the centroid-to-midpoint part or vice versa, I will assume the following for the ambiguous ones:

- In problem 1: "YN" is meant to be the segment from the centroid to the midpoint N, i.e., CN. Given UN=10, then CN = (1/3)*10 = 10/3.
- In problem 2: "JQ" is meant to be the segment from the centroid to the midpoint K, i.e., CK. Given KQ=16 (full median), then CK = (1/3)*16 = 16/3.
- In problem 4: "JH" is meant to be the full median from J, but since we have CH=17 (vertex to centroid), and if we assume the median from J is similar, but that's not valid. Perhaps "JH" is meant to be "CM" or something else. Given CH=17 (vertex to centroid), then the full median HM = (3/2)*17 = 25.5, and CM = 8.5. But the question asks for JH. Perhaps it's a typo and should be "HM" or "CM". I'll assume it's "CM" = 8.5.
- In problem 6: "RZ" is likely meant to be "RO" = 14 (full median from R), and "OZ" is meant to be "CO" = (1/3)*14 = 14/3.
- In problem 7: "TA" = 12 (full median), and "JA" is meant to be "CA" = (1/3)*12 = 4.

This is speculative, but it's the only way to proceed.

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## Final Answers (with assumptions):

1) YN = 10/3 (assuming YN means CN, the centroid to midpoint part)
2) JQ = 16/3 (assuming JQ means CK, the centroid to midpoint part)
3) CS = 12
4) JH = 8.5 (assuming JH means CM, the centroid to midpoint part, given CH=17)
5) CY = 17
6) OZ = 14/3 (assuming OZ means CO, the centroid to midpoint part, given RO=14)
7) JA = 4 (assuming JA means CA, the centroid to midpoint part, given TA=12)
8) CH = 8

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## Boxed Answers:

To provide the final answer as requested:

1) \(\boxed{\dfrac{10}{3}}\)

2) \(\boxed{\dfrac{16}{3}}\)

3) \(\boxed{12}\)

4) \(\boxed{8.5}\)

5) \(\boxed{17}\)

6) \(\boxed{\dfrac{14}{3}}\)

7) \(\boxed{4}\)

8) \(\boxed{8}\)

Note: These answers are based on reasonable assumptions due to potential labeling errors in the worksheet. In a real classroom setting, clarification would be needed for problems 1, 2, 4, 6, and 7.