Examples on Median of a Triangle

Some examples on Median of a Triangle are,

Example 1: In triangle DEF, if DE = 7 cm, DF = 9 cm, and EF = 10 cm, calculate the length of the median from vertex D to side EF.

Solution:

To find length of median from vertex D to side EF in triangle DEF, we can use formula for length of a median in terms of lengths of sides of triangle. Let’s denote length of median as (m).

Given,

• DE = 7
• DF = 9
• EF = 10

Use Formula:

m² = [Tex]\frac{2(DF^2 + DE^2) – EF^2}{4}[/Tex]

Put given values:

m² = [Tex]\frac{2(9^2 + 7^2) – 10^2}{4}[/Tex]

⇒ m² = [Tex]\frac{2(81 + 49) – 100}{4}[/Tex]

⇒ m² = [Tex]\frac{2(130) – 100}{4}[/Tex]

⇒ m² = (260 – 100)/4

⇒ m² = 160/40

⇒ m² = 40

⇒ m = √40

⇒ m = 2√10

So, length of median from vertex D to side EF is 2√10 cm.

Example 2: Determine the length of median from vertex C to side AB in a triangle where AC = 12 cm, BC = 9 cm, and AB = 15 cm.

Solution:

To find length of median from vertex C to side AB in given triangle, use formula for length of median:

m² = [Tex]\frac{2(BC^2 + AC^2) – AB^2}{4}[/Tex]

Given,

• AC = 12
• BC = 9
• AB = 15

Use Formula:

m² = [Tex]\frac{2(9^2 + 12^2) – 15^2}{4}[/Tex]

⇒ m² = [Tex]\frac{2(81 + 144) – 225}{4}[/Tex]

⇒ m² = [Tex]\frac{2(225) – 225}{4}[/Tex]

⇒ m² = [Tex]\frac{450 – 225}{4}[/Tex]

⇒ m² = 225/4

⇒ m = [Tex]\sqrt{\frac{225}{4}}[/Tex]

⇒ m = 15/2

⇒ m = 7.5

So, length of median from vertex C to side AB is (7.5) cm

Example 3: Triangle UVW has vertices U (3, 5), V (9, 5), and W (6, 1). Find length of median from vertex U to side VW.

Solution:

To find length of median from vertex U to side VW in triangle UVW, First determine midpoint of side VW, and then use distance formula to find length of median.

Given,

• Coordinates of Vertex = V(9, 5)
• Coordinates of Vertex = W(6, 1)

1. Find midpoint of side VW

Midpoint formula is given by:

[Tex]\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) [/Tex]

Using coordinates of V and W:

[Tex]\text{Midpoint}(VW) = \left( \frac{9 + 6}{2}, \frac{5 + 1}{2} \right)[/Tex]

[Tex]=~\left( \frac{15}{2}, \frac{6}{2} \right)[/Tex]

[Tex]=~\left( \frac{15}{2}, 3 \right) [/Tex]

So, the midpoint of side VW is [Tex]\left( \frac{15}{2}, 3 \right)[/Tex]

2. Now, use distance formula to find length of median from vertex U to midpoint of side VW

Distance Formula is:

[Tex]d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} [/Tex]

Given,

• Coordinates of Vertex = U(3, 5)

[Tex]\text{Midpoint}(VW) = \left( \frac{15}{2}, 3 \right)[/Tex]

Using distance formula:

d = [Tex]\sqrt{\left( \frac{15}{2} – 3 \right)^2 + (3 – 5)^2} [/Tex]

⇒ d = [Tex]\sqrt{\left( \frac{15}{2} – 3 \right)^2 + (-2)^2}[/Tex]

⇒ d = [Tex]\sqrt{\left( \frac{15}{2} – \frac{6}{2} \right)^2 + 4}[/Tex]

⇒ d = [Tex]\sqrt{\left( \frac{9}{2} \right)^2 + 4}[/Tex]

⇒ d = [Tex]\sqrt{\frac{81}{4} + 4} [/Tex]

⇒ d = [Tex]\sqrt{\frac{81 + 16}{4}}[/Tex]

⇒ d = [Tex]\sqrt{\frac{97}{4}}[/Tex]

⇒ d = √(97)/2

So, length of median from vertex U to side VW is √(97)/2

Median of a Triangle is a line segment that joins a vertex of a triangle to the midpoint of the opposite side. A median divides the joining into two equal parts. Each triangle has three medians, one originating from each vertex. These medians intersect at a point called the centroid, which lies within the triangle.

In this article, we will learn about, Median of Triangle Definition, Properties of Median of Triangle, Examples related to Median of Triangle, and others in detail.