Triangle Medians worksheet with eight problems involving finding segment lengths in triangles with medians.
Worksheet titled "Triangle Medians" with eight problems, each showing a triangle with its three medians drawn, asking to find specific segment lengths based on given values.
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Triangle Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Triangle Worksheets
Let’s solve each problem one by one. Remember: In a triangle, the medians intersect at a point called the centroid, and the centroid divides each median into two parts — the part from the vertex to the centroid is twice as long as the part from the centroid to the midpoint of the opposite side.
So, if you have a median split by the centroid C:
- The whole median = 3 parts
- From vertex to C = 2 parts
- From C to midpoint = 1 part
That means:
- If you know the shorter part (from C to midpoint), multiply by 3 to get the full median.
- If you know the longer part (vertex to C), multiply by 3/2 to get the full median.
- If you know the full median, divide by 3 to get the short part, or multiply by 2/3 to get the long part.
Now let’s go question by question.
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1) Find YN if UN = 10.
UN is the entire median from U to N. Y is the midpoint of UN? Wait — look at the diagram: In triangle QUN, medians are drawn from each vertex to the midpoint of the opposite side. So:
- Median from Q goes to Y (midpoint of UN)
- Median from U goes to V (midpoint of QN)
- Median from N goes to O (midpoint of QU)
They meet at centroid C.
We’re told UN = 10. But UN is a SIDE of the triangle, not a median! Wait — no, in the diagram, UN is labeled as a side, but the median from Q hits UN at Y. So Y is the midpoint of UN → so UY = YN.
If UN = 10, and Y is the midpoint, then YN = half of UN → YN = 5.
Wait — but the problem says “Find YN if UN = 10.” And in the diagram, Y is on UN, and it’s where the median from Q lands → so yes, Y is midpoint → YN = 5.
But hold on — maybe I misread. Let me check again.
Actually, looking at the first triangle: vertices Q, U, N. Medians: from Q to Y (on UN), from U to V (on QN), from N to O (on QU). They meet at C.
So UN is a side. Y is midpoint of UN → so if UN = 10, then YN = 5.
✔ Answer for #1: 5
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2) Find JQ if KQ = 16.
Triangle DJQ. Medians: from D to K (on JQ), from J to M (on DQ), from Q to E (on DJ). Meet at C.
K is on JQ, and DK is the median → so K is midpoint of JQ.
Therefore, JK = KQ.
Given KQ = 16 → so JK = 16 → JQ = JK + KQ = 16 + 16 = 32.
✔ Answer for #2: 32
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3) SP = 18. Find CS.
Triangle USY. Medians: from U to K (on SY), from S to P (on UY), from Y to Q (on US). Meet at C.
SP is the median from S to P (P is midpoint of UY).
C is centroid → so SC : CP = 2 : 1.
Total SP = 18 → divided into 3 parts → each part = 6.
CS is the part from C to S → that’s 2 parts → 2 × 6 = 12.
Wait — CS is from C to S? Yes, since S is the vertex, C is centroid, P is midpoint → so SC = 2/3 of SP.
SP = 18 → SC = (2/3)*18 = 12.
✔ Answer for #3: 12
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4) CH = 17. Find JH.
Triangle JPS. Medians: from P to M (on JS), from J to H (on PS), from S to B (on JP). Meet at C.
JH is the median from J to H (H is midpoint of PS).
C is centroid → so JC : CH = 2 : 1.
Given CH = 17 → that’s the smaller part (from centroid to midpoint).
So total JH = JC + CH = 2 parts + 1 part = 3 parts.
Since CH = 1 part = 17 → JH = 3 × 17 = 51.
✔ Answer for #4: 51
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5) BY = 51. Find CY.
Triangle PBS. Medians: from P to Z (on BS), from B to Y (on PS), from S to F (on PB). Meet at C.
BY is the median from B to Y (Y is midpoint of PS).
C is centroid → so BC : CY = 2 : 1.
Total BY = 51 → divided into 3 parts → each part = 17.
CY is the part from C to Y → that’s 1 part → 17.
✔ Answer for #5: 17
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6) Find OZ if RZ = 14.
Triangle TRZ. Medians: from T to O (on RZ), from R to I (on TZ), from Z to Y (on TR). Meet at C.
RZ is a side. O is on RZ, and TO is the median → so O is midpoint of RZ.
Therefore, RO = OZ.
Given RZ = 14 → so OZ = half of 14 = 7.
✔ Answer for #6: 7
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7) Find JA if TA = 12.
Triangle LJA. Medians: from L to T (on JA), from J to Z (on LA), from A to Q (on LJ). Meet at C.
TA is part of the median from L to T? Wait — TA is from T to A. T is on JA, and LT is the median → so T is midpoint of JA.
Therefore, JT = TA.
Given TA = 12 → so JT = 12 → JA = JT + TA = 12 + 12 = 24.
✔ Answer for #7: 24
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8) HA = 12. Find CH.
Triangle DHL. Medians: from D to E (on HL), from H to A (on DL), from L to P (on DH). Meet at C.
HA is the median from H to A (A is midpoint of DL).
C is centroid → so HC : CA = 2 : 1.
Total HA = 12 → divided into 3 parts → each part = 4.
CH is the part from C to H → that’s 2 parts → 2 × 4 = 8.
Wait — CH is from C to H? Yes, H is vertex, C is centroid, A is midpoint → so HC = 2/3 of HA.
HA = 12 → HC = (2/3)*12 = 8.
✔ Answer for #8: 8
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Final Answer:
1) 5
2) 32
3) 12
4) 51
5) 17
6) 7
7) 24
8) 8
So, if you have a median split by the centroid C:
- The whole median = 3 parts
- From vertex to C = 2 parts
- From C to midpoint = 1 part
That means:
- If you know the shorter part (from C to midpoint), multiply by 3 to get the full median.
- If you know the longer part (vertex to C), multiply by 3/2 to get the full median.
- If you know the full median, divide by 3 to get the short part, or multiply by 2/3 to get the long part.
Now let’s go question by question.
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1) Find YN if UN = 10.
UN is the entire median from U to N. Y is the midpoint of UN? Wait — look at the diagram: In triangle QUN, medians are drawn from each vertex to the midpoint of the opposite side. So:
- Median from Q goes to Y (midpoint of UN)
- Median from U goes to V (midpoint of QN)
- Median from N goes to O (midpoint of QU)
They meet at centroid C.
We’re told UN = 10. But UN is a SIDE of the triangle, not a median! Wait — no, in the diagram, UN is labeled as a side, but the median from Q hits UN at Y. So Y is the midpoint of UN → so UY = YN.
If UN = 10, and Y is the midpoint, then YN = half of UN → YN = 5.
Wait — but the problem says “Find YN if UN = 10.” And in the diagram, Y is on UN, and it’s where the median from Q lands → so yes, Y is midpoint → YN = 5.
But hold on — maybe I misread. Let me check again.
Actually, looking at the first triangle: vertices Q, U, N. Medians: from Q to Y (on UN), from U to V (on QN), from N to O (on QU). They meet at C.
So UN is a side. Y is midpoint of UN → so if UN = 10, then YN = 5.
✔ Answer for #1: 5
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2) Find JQ if KQ = 16.
Triangle DJQ. Medians: from D to K (on JQ), from J to M (on DQ), from Q to E (on DJ). Meet at C.
K is on JQ, and DK is the median → so K is midpoint of JQ.
Therefore, JK = KQ.
Given KQ = 16 → so JK = 16 → JQ = JK + KQ = 16 + 16 = 32.
✔ Answer for #2: 32
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3) SP = 18. Find CS.
Triangle USY. Medians: from U to K (on SY), from S to P (on UY), from Y to Q (on US). Meet at C.
SP is the median from S to P (P is midpoint of UY).
C is centroid → so SC : CP = 2 : 1.
Total SP = 18 → divided into 3 parts → each part = 6.
CS is the part from C to S → that’s 2 parts → 2 × 6 = 12.
Wait — CS is from C to S? Yes, since S is the vertex, C is centroid, P is midpoint → so SC = 2/3 of SP.
SP = 18 → SC = (2/3)*18 = 12.
✔ Answer for #3: 12
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4) CH = 17. Find JH.
Triangle JPS. Medians: from P to M (on JS), from J to H (on PS), from S to B (on JP). Meet at C.
JH is the median from J to H (H is midpoint of PS).
C is centroid → so JC : CH = 2 : 1.
Given CH = 17 → that’s the smaller part (from centroid to midpoint).
So total JH = JC + CH = 2 parts + 1 part = 3 parts.
Since CH = 1 part = 17 → JH = 3 × 17 = 51.
✔ Answer for #4: 51
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5) BY = 51. Find CY.
Triangle PBS. Medians: from P to Z (on BS), from B to Y (on PS), from S to F (on PB). Meet at C.
BY is the median from B to Y (Y is midpoint of PS).
C is centroid → so BC : CY = 2 : 1.
Total BY = 51 → divided into 3 parts → each part = 17.
CY is the part from C to Y → that’s 1 part → 17.
✔ Answer for #5: 17
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6) Find OZ if RZ = 14.
Triangle TRZ. Medians: from T to O (on RZ), from R to I (on TZ), from Z to Y (on TR). Meet at C.
RZ is a side. O is on RZ, and TO is the median → so O is midpoint of RZ.
Therefore, RO = OZ.
Given RZ = 14 → so OZ = half of 14 = 7.
✔ Answer for #6: 7
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7) Find JA if TA = 12.
Triangle LJA. Medians: from L to T (on JA), from J to Z (on LA), from A to Q (on LJ). Meet at C.
TA is part of the median from L to T? Wait — TA is from T to A. T is on JA, and LT is the median → so T is midpoint of JA.
Therefore, JT = TA.
Given TA = 12 → so JT = 12 → JA = JT + TA = 12 + 12 = 24.
✔ Answer for #7: 24
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8) HA = 12. Find CH.
Triangle DHL. Medians: from D to E (on HL), from H to A (on DL), from L to P (on DH). Meet at C.
HA is the median from H to A (A is midpoint of DL).
C is centroid → so HC : CA = 2 : 1.
Total HA = 12 → divided into 3 parts → each part = 4.
CH is the part from C to H → that’s 2 parts → 2 × 4 = 8.
Wait — CH is from C to H? Yes, H is vertex, C is centroid, A is midpoint → so HC = 2/3 of HA.
HA = 12 → HC = (2/3)*12 = 8.
✔ Answer for #8: 8
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Final Answer:
1) 5
2) 32
3) 12
4) 51
5) 17
6) 7
7) 24
8) 8
Parent Tip: Review the logic above to help your child master the concept of medians worksheet.
