How to Find the Length of a Midsegment in a Triangle: Step-by-Step Guide and Examples

Midsegment

The midsegment is a line segment that connects the midpoints of two sides of a triangle

The midsegment is a line segment that connects the midpoints of two sides of a triangle. It is also known as the midline. In a triangle, there are three midsegments that can be formed.

To find the length of a midsegment, you need to find the midpoint of each side and then calculate the distance between those midpoints.

Let’s take an example to demonstrate this.

Consider a triangle ABC with side lengths AB = 8 cm, BC = 6 cm, and AC = 10 cm.

To find the midsegment connecting the midpoints of AB and BC, follow these steps:

Step 1: Find the midpoint of AB:
The midpoint formula states that the x-coordinate of the midpoint (M) is the average of the x-coordinates of the endpoints (A and B), and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.

In this case, A = (0, 0) and B = (8, 0).
The x-coordinate of the midpoint = (0 + 8) / 2 = 4
The y-coordinate of the midpoint = (0 + 0) / 2 = 0

So, the midpoint of AB, M1, is (4, 0).

Step 2: Find the midpoint of BC:
In this case, B = (8, 0) and C = (6, 8).
The x-coordinate of the midpoint = (8 + 6) / 2 = 7
The y-coordinate of the midpoint = (0 + 8) / 2 = 4

So, the midpoint of BC, M2, is (7, 4).

Step 3: Calculate the distance between the midpoints:
Using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:

Distance = sqrt((x2 – x1)^2 + (y2 – y1)^2)

In this case, the coordinates of M1 are (4, 0) and the coordinates of M2 are (7, 4).

Distance = sqrt((7 – 4)^2 + (4 – 0)^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5

Therefore, the length of the midsegment connecting the midpoints of AB and BC is 5 cm.

You can follow the same steps to find the length of other midsegments in the triangle. Remember to calculate the midpoint of the two sides and then find the distance between those midpoints using the distance formula.

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