THE LENGTHS OF THE MEDIANS OF A TRIANGLE

Given the lengths of the sides of a triangle, how to find the lengths of the medians? There are several formulae for determining the lengths of the medians. They are easily derived by applying Stewart’s Theorem.

Stewart’s Theorem

Let ABC be any triangle. Suppose that the point D lies on the line segment $\overline{AB}$ such that AD = c₁ and DB = c₂. Let BC = a, AC = b, AB = c. (See Figure 1.)

Figure 1

Let CD = $\ell$ . Then,

$\ell^2 c = a^2 c_1 + b^2 c_2 - c_1 c_2 c$ …………………………………………………………………………………………………………………………………………………….. (SW)

If $\overline{CD}$ is the median of ∆ABC that passes through the vertex C, then $c_1 = c_2 = \frac{c}{2}$ . Substituting c₁ and c₂ into (SW), we get:

$\ell^2 c = a^2 \cdot \frac{c}{2} + b^2 \cdot \frac{c}{2} - \frac{c}{2} \cdot \frac{c}{2} \cdot c$

$\ell^2 c = \frac{1}{2} a^2 c + \frac{1}{2} b^2 c - \frac{1}{4} c^3$

Multiply both sides of the equation by $\frac{1}{c}$ . This yields:

$\ell^2 = \frac{1}{2} a^2 + \frac{1}{2} b^2 - \frac{1}{4} c^2$ ……………………………………………………………………………………………………………………………………………………………………… (*)

Thus, the length of the median is equal to the square root of the expression on the right-hand side of (*), that is:

$\ell = \sqrt{\frac{1}{2} a^2 + \frac{1}{2} b^2 - \frac{1}{4} c^2}$

By simplifying the square root above, we then have:

$\ell = \frac{1}{2} \sqrt{2a^2+2b^2-c^2}$ ………………………………………………………………………………………………………………………………………………………………. (**)

Now, let:

the length of the median passing through vertex C = z_c ,

the length of the median passing through vertex B = z_b , and

the length of the median passing through vertex A = z_a

Then, by applying (**) we can conclude that the lengths of the medians are as follows.

$z_c = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2}$ …………………………………………………………………………………………………………………………………………………………………… (1)

$z_b = \frac{1}{2} \sqrt{2a^2 + 2c^2 - b^2}$ …………………………………………………………………………………………………………………………………………………………………. (2)

$z_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}$ ……………………………………………………………………………………………………………………………………………………………….. (3)

(See Figure 2.)

Figure 2

(In Figure 2, a = BC, b = AC, and c = AB.)

Example

Given ΔABC with AB = 12 cm, BC = 9 cm, and AC = 5 cm, find the lengths of its medians.

Answer

In this example, a = 9 cm, b = 5 cm, and c = 12 cm.

The length of the median passing through vertex A is z_a. By applying (3), we have:

$z_a = \frac{1}{2} \sqrt{2 \cdot 5^2 + 2 \cdot 12^2 - 9^2} \:cm$

$z_a = \frac{\sqrt{257}}{2} \:cm$

The length of the median passing through vertex B is z_b. By applying (2), we have:

$z_b = \frac{1}{2} \sqrt{2 \cdot 9^2 + 2 \cdot 12^2 - 5^2} \:cm$

$z_b = \frac{5 \sqrt{17}}{2} \: cm$

The length of the median passing through vertex C is z_c. By applying (1), we have:

$z_c = \frac{1}{2} \sqrt{2 \cdot 9^2 + 2 \cdot 5^2 - 12^2} \:cm$

$z_c = \sqrt{17} \:cm$

When solving the problem above, z_a, z_b, and z_c are calculated using (3), (2), and (1). However, if any two of z_a, z_b, and z_c have been calculated, the length of the other median can also be computed using the following formula.

4(z_a² + z_b² + z_c²) = 3(a² + b² + c²) ………………………………………………………………………………………………………………………………………. (**)

The formula (**) shows the ratio of the sum of the squares of the lengths of sides of the triangle to the sum of the squares of the lengths of the medians:

(a² + b² + c²) : (z_a² + z_b² + z_c²) = 4 : 3 ……………………………………………………………………………………………….. (***)

In the example above, suppose that we have obtained the values of z_a and z_b. To find z_c, we can use (***). We proceed as follows.

$\frac{a^2 + b^2 + c^2}{{z_a}^2 + {z_b}^2 + {z_c}^2} = \frac{4}{3}$ ………………………………………………………………………………………………………………………………………………………………………………… (o)

Substitute a = 9, b = 5, c = 12, $z_a = \frac{\sqrt{257}}{2}$ , and $z_b = \frac{5 \sqrt{17}}{2}$ into (o). This results in:

$\frac{9^2 + 5^2 + 12^2}{(\frac{1}{2} \sqrt{257})^2 + (\frac{5}{2} \sqrt{17})^2 + {z_c}^2} = \frac{4}{3}$

It is easy to check that $z_c = \sqrt{17}$ satisfies the equation. (We have ignored the negative root of the equation because the length of the median cannot be negative.) Thus, the length of the median that passes through vertex C is $\sqrt{17} \: cm$ .

On 17/04/2022

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