Skills Practice DATE PERIOD Medians and Altitudes of Triangles In \triangle PQR, NQ = 6, RK = 3, and PK = 4. Find each measure.

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Skills Practice DATE PERIOD Medians and Altitudes of Triangles In \triangle PQR, NQ = 6, RK = 3, and PK = 4. Find each measure. L 1. KM 2. KQ K M P 4 2 N R 3. LK 4. LR 1.5 4.5 5. NK 6. PM 2 6 In \triangle STR, H is the centroid, EH = 6, DH = 4, and SM = 24. Find each measure. S D 8. HM 7. SH 8 16 9. TH 10. HR 12 8 11. TD 12. ER

The Correct Answer and Explanation is:

Let’s analyze and solve the worksheet titled “Skills Practice: Medians and Altitudes of Triangles” step-by-step, followed by a 300-word explanation of the geometry concepts involved.

Given:

For triangle ΔPQR:

  • NQ = 6, RK = 3, PK = 4.
  • M, K, and N are midpoints, so segments like KM, KQ, etc., involve medians.

Answers:

  1. KM = 2
  2. KQ = 4
  3. LK = 1.5
  4. LR = 4.5
  5. NK = 2
  6. PM = 6

For triangle ΔSTR:

  • H is the centroid.
  • EH = 6, DH = 4, SM = 24.

Answers:

  1. SH = 16
  2. HM = 8
  3. TH = 8
  4. HR = 12
  5. TD = 12
  6. ER = 16

Explanation

This worksheet explores medians and centroids in triangles. A median connects a triangle’s vertex to the midpoint of the opposite side. All three medians intersect at the centroid, which has special properties. The centroid divides each median into a 2:1 ratio, with the longer segment always between the vertex and the centroid.

In triangle ΔPQR, you’re often asked to find lengths of segments that are part of medians. For instance, if RK = 3, and K is the midpoint, then the full median must be twice that value when extended beyond the midpoint. Similarly, segment relationships can be determined using midpoint and triangle congruence rules.

In triangle ΔSTR, the centroid H plays a key role. A key property of centroids is:

  • The distance from the vertex to the centroid is 2/3 of the entire median.
  • The distance from the centroid to the midpoint of the opposite side is 1/3 of the median.

For example, if EH = 6 (from midpoint to centroid), then the total median SE = 18, making SH = 12 (since 6 × 2 = 12). If DH = 4 (centroid to midpoint), then SD = 12, making TH = 8, and so on.

Understanding the 2:1 centroid rule and applying midpoint logic lets you find unknown measures efficiently in triangle diagrams involving medians and centroids.

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