Solving for Median Length in Triangle ABC

In the context of triangle geometry, finding the length of a median can be a direct application of various theorems. In this article, we will demonstrate how to solve for the length of median MA in a triangle with given side lengths. The triangle ABC has sides BC 8, CA 4, and AB 6.

Method 1: Using Heron's Formula

Let M denote the midpoint of side BC. We are tasked with finding the length of AM.

First, recognize that the area of triangles AMC and AMB are equal due to their shared height and equal bases MC and MB. Using Heron's formula, we start by calculating the semi-perimeter s for triangle BCM, which is the same for both triangles AMC and AMB.

Let the length of AM be denoted as 2x, and apply Heron's formula as follows:

s (8 4 2x) / 2 6 x

Heron's formula states: Area √[s(s - 8)(s - 4)(s - 2x)]

Setting these two triangles equal in area:

x1x-1x5-5 x2x4-4

Thus, solving for x:

x ±√(5/2)

Since x must represent a positive length:

x √(5/2)

Therefore, the length of AM is:

AM 2x √10

Method 2: Using the Perpendicular from A to BC

Draw AN perpendicular to BC, with MN x. Using Heron's formula to find the area of triangle ABC, we have:

Area √(99 - 89 - 49 - 6) √(9, 1, 5, 3) √135

From the right triangles ABN and ACN:

AN2 AB2 - BN2 AC2 - CN2

Since BN and CN are not directly given, we solve for x:

AN2 (62 - 42) - 4x2 (42 - 4) - x2

This simplifies to a linear form, which then helps to determine:

x 5/4

Now, the area of triangle ABC is 8AN/2 √135, so AN √135/4 135/16.

From triangle AMN:

AM2 AN2 - MN2 135/16 - 25/16 10

Thus, AM √10 ≈ 3.1623.

Method 3: Applying Apollonius' Theorem

According to Apollonius' theorem, the formula for the median of a triangle with sides of lengths a, b, and c, and the median AM is:

b2 c2 2a2 2AM2

Substituting the known values:

42 62 2(82) 2AM2

This simplifies to:

16 36 128 2AM2

Solving for AM:

AM2 (42 - 32) / 2 10

Thus, AM √10 ≈ 3.1623.

Note: It is worth noting that in a typical notation, the length of a line segment, such as AB, is denoted as the absolute value of the endpoints; e.g., AB