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Chapter 5: Problem 37

If \(A B>A C>B C\) in \(\triangle A B C\) and \(\overline{A M}, \overline{B N},\) and \(\overline{C O}\) are the medians of the triangle, list \(A M, B N,\) and \(C O\) in order from least to greatest.

Short Answer

Expert verified
In order of length: \( \overline{CO}, \overline{BN}, \overline{AM} \).

Step by step solution

01

Understand the Problem

We are given a triangle \( \triangle ABC \) with the inequality \( AB > AC > BC \). We need to arrange the medians \( \overline{AM}, \overline{BN}, \) and \( \overline{CO} \) in order from least to greatest.
02

Recall Properties of Medians and Triangles

In any triangle, the length of a median connecting a vertex to the midpoint of the opposite side is affected by the lengths of the triangle's sides. Specifically, the longer the opposing side, the shorter the median from that vertex.
03

Analyze Median Lengths Relative to Side Lengths

Since \( AB > AC > BC \), \( \overline{CO} \) is the median from the longest side \( AB \), making \( \overline{CO} \) the shortest median. Conversely, \( \overline{AM} \) is opposed by the shortest side \( BC \), making it the longest median. \( \overline{BN} \) is the median from the side \( AC \), which is neither the longest nor the shortest, thus it has a median of intermediate length.
04

Order the Medians

Based on the previous step, order the medians from shortest to longest: \( \overline{CO} < \overline{BN} < \overline{AM} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Medians of a Triangle
Medians in a triangle have a special characteristic. Each median divides the triangle into two smaller triangles of equal area. This is because a median is a line segment that connects a vertex to the midpoint of the opposite side. Each triangle has three medians, one from each vertex.
The interesting thing about medians is their relationship with the triangle's sides. If you consider the medians' lengths, they vary depending on the length of the triangle's sides.
  • The median from the longest side is the shortest among the medians.
  • The median from the shortest side is the longest.
  • Medians not from the longest or shortest sides are of intermediate length.
This relationship comes handy when you are to order the medians by length, as done in our exercise.
Triangle Inequality Theorem
Understanding triangle inequalities is crucial for geometry problem solving, particularly when examining medians. The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
This ensures that you can form a triangle with the given side lengths and avoids the scenario of having a 'degenerate' triangle where one side is merely equal to the sum of the other two.
When applied practically, such as in the exercise mentioned, the theorem facilitates the understanding of the relationship between side lengths and median lengths, helping to solve problems by understanding which sides relate to which medians.
Geometry Problem Solving
Solving geometry problems that involve triangles and medians requires a mix of understanding properties and using logical reasoning. Begin by examining the properties of the triangle and identifying any given inequalities.
In the exercise we did, knowing the side lengths allowed us to determine median lengths. And then we ordered those medians based on the inequality relationship between the triangle's sides.
To effectively solve similar problems:
  • Analyze given information carefully, noting all side relationships.
  • Use known theorems, like the Triangle Inequality Theorem, to check the triangle's validity.
  • Relate medians or other segments within triangles to these sides, applying logical reasoning to deduce relative lengths.
By continually practicing, you enhance your skills in geometry problem solving, making these processes feel intuitive over time.

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