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Midsegments, Medians & Altitudes - Worksheet 9 - 10th Grade

Hard

Apply properties of triangle midsegments, medians, and altitudes (Hard)

23 problems · 20-25 minutes · Skills: triangles, geometric properties, geometry

About This Worksheet

This hard-level worksheet contains 23 problems designed for 10th Grade students practicing midsegments, medians & altitudes. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

The special segments of a triangle reveal deep structural properties — centers of gravity, circumscribed circles, and inscribed circles. These concepts connect to physics, engineering design, and later to coordinate geometry proofs where proving a point is a midpoint or the centroid requires algebraic calculation.

  • Apply the Triangle Midsegment Theorem to find lengths and prove parallelism
  • Locate and use the centroid as the intersection of the three medians
  • Distinguish among the orthocenter, circumcenter, incenter, and centroid of a triangle
  • Use the centroid theorem to find segment lengths when given a median length
  • Construct the circumscribed and inscribed circles of a triangle using the circumcenter and incenter

How to Solve: Worked Example

Example Problem: The centroid divides a median 2:1. If the shorter piece is 10, find the full median.
Solution Steps:
  1. The centroid splits the median into parts with ratio 2:1.
  2. The shorter piece is 1 part = 10, so the longer piece is 2 parts = 20.
  3. The whole median is 10 + 20 = 30.
Answer: 30
Tip: The longer piece always runs from the vertex to the centroid — it is two-thirds of the whole median.
1
The centroid divides a median in a 2:1 ratio. If the shorter piece is 30, find the full median length.
2
The centroid divides a median in a 2:1 ratio. If the shorter piece is 8, find the full median length.
3
The centroid divides a median in a 2:1 ratio. If the shorter piece is 10, find the full median length.
4
The centroid divides a median in a 2:1 ratio. If the shorter piece is 29, find the full median length.
5
The centroid divides a median in a 2:1 ratio. If the shorter piece is 36, find the full median length.
6
The centroid divides a median in a 2:1 ratio. If the shorter piece is 31, find the full median length.
7
The centroid divides a median in a 2:1 ratio. If the shorter piece is 15, find the full median length.
8
The centroid divides a median in a 2:1 ratio. If the shorter piece is 32, find the full median length.
9
The centroid divides a median in a 2:1 ratio. If the shorter piece is 2, find the full median length.
10
The centroid divides a median in a 2:1 ratio. If the shorter piece is 29, find the full median length.
11
The centroid divides a median in a 2:1 ratio. If the shorter piece is 21, find the full median length.
12
The centroid divides a median in a 2:1 ratio. If the shorter piece is 33, find the full median length.
13
The centroid divides a median in a 2:1 ratio. If the shorter piece is 7, find the full median length.
14
The centroid divides a median in a 2:1 ratio. If the shorter piece is 31, find the full median length.
15
The centroid divides a median in a 2:1 ratio. If the shorter piece is 7, find the full median length.
16
The centroid divides a median in a 2:1 ratio. If the shorter piece is 5, find the full median length.
17
The centroid divides a median in a 2:1 ratio. If the shorter piece is 35, find the full median length.
18
The centroid divides a median in a 2:1 ratio. If the shorter piece is 17, find the full median length.
19
The centroid divides a median in a 2:1 ratio. If the shorter piece is 35, find the full median length.
20
The centroid divides a median in a 2:1 ratio. If the shorter piece is 25, find the full median length.
21
The centroid divides a median in a 2:1 ratio. If the shorter piece is 3, find the full median length.
22
The centroid divides a median in a 2:1 ratio. If the shorter piece is 24, find the full median length.
23
The centroid divides a median in a 2:1 ratio. If the shorter piece is 24, find the full median length.

Answer Key

1. 902. 243. 304. 875. 1086. 937. 458. 969. 610. 8711. 6312. 9913. 2114. 9315. 2116. 1517. 10518. 5119. 10520. 7521. 922. 7223. 72

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What skills does this triangles worksheet cover?

This worksheet focuses on triangles, geometric properties, geometry. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

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Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 20-25 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.