Introduction

When dealing with a triangular lot with sides of 125m, 175m, and 190m, it can be useful to divide the lot using a line that bisects the longest side (190m) and is drawn from the opposite vertex. This line, commonly referred to as the median, can be calculated using specific geometric principles. This article will explore the methods to find the length of this dividing line, including the use of Stewart's theorem, Apollonius' theorem, and the concept of medians in triangles.

Understanding Medians in Triangles

A median of a triangle is a line segment which connects a vertex to the midpoint of the opposite side. In this case, we need to find the length of the median from the vertex opposite the longest side of 190m.

Stewart's Theorem

Stewart's theorem is useful for finding the length of a median in a triangle. Let's denote the sides of the triangle as follows: sides (a) and (b) are 175m and 125m respectively, and side (c) is the longest side, which is 190m. The median from the vertex opposite side (c) will divide side (c) into two equal segments, each of length (95m).

To find the length of the median, we will use Stewart's formula:

[d^2 frac{2b^2c 2a^2d - a^2b^2}{4d}]

Let's substitute the values into the formula:

[d^2 frac{2 times 125^2 times 190 2 times 175^2 times 95 - 190^2 times 175}{4 times 95}]

After calculating the terms, we get:

[d approx 118.74 , text{meters}]

Using Apollonius' Theorem

Apollonius' theorem can also be used to find the length of the median in a triangle. The theorem states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the square of the median to the first side. The formula is:

[c^2 2b^2 2a^2 - 4x^2]

Where (c 190m), (b 125m), (a 175m), and (x) is the length of the median. Solving for (x), we get:

[x approx 118.74 , text{meters}]

Calculating with the Area Method

We can also find the length of the median using the method of dividing the triangle into two smaller triangles and calculating their areas. By dividing the triangle into two triangles using the median, we can equate their areas to find the length of the median.

The semiperimeter (s) of the triangle is given by:

[s frac{125 175 190}{2} 245 , text{meters}]

The area of the triangle can be found using Heron's formula, and by dividing the triangle into two smaller triangles, we can solve for the length of the median. After solving, we find that the length of the median is approximately 118.74 meters.

Conclusion

In conclusion, the length of the line bisecting the longest side of a triangular lot, drawn from the opposite vertex, can be calculated using Stewart's theorem, Apollonius' theorem, or by dividing the triangle into two smaller triangles. The length of this dividing line is approximately 118.74 meters.

Keywords: triangular lot, median of a triangle, Stewart's theorem