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Step-by-Step Solution
Step 1: Find the points of intersection
We have three lines:
x - y + 1 = 0
x - 2y + 3 = 0
2x - 5y + 11 = 0
Their points of intersection are given as midpoints of the sides of the triangle \\Delta ABC . From the solution, these points are:
D = (1,\\,2)
E = (7,\\,5)
F = (2,\\,3)
Step 2: Calculate the area of triangle formed by these midpoints
To find the area of \\Delta DEF , we can use the matrix (determinant) formula for the area of a triangle with vertices (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) :
\\Delta = \\frac{1}{2} \\left|
\\begin{matrix}
x_1 & y_1 & 1 \\\\
x_2 & y_2 & 1 \\\\
x_3 & y_3 & 1
\\end{matrix}
\\right|
Substitute D = (1, 2) , E = (7, 5) , and F = (2, 3) :
\\Delta DEF
= \\frac{1}{2} \\left|
\\begin{matrix}
1 & 2 & 1 \\\\
7 & 5 & 1 \\\\
2 & 3 & 1
\\end{matrix}
\\right|
Compute the determinant:
\\begin{aligned}
\\Delta DEF
&= \\frac{1}{2} \\bigl[
1 \\times (5 - 3)
\\;-\\; 2 \\times (7 - 2)
\\;+\\; 1 \\times (21 - 10)
\\bigr] \\\\
&= \\frac{1}{2} \\bigl[ (1 \\times 2) - (2 \\times 5) + (1 \\times 11) \\bigr] \\\\
&= \\frac{1}{2} \\bigl[ 2 - 10 + 11 \\bigr]
= \\frac{1}{2} \\times 3
= \\frac{3}{2}.
\\end{aligned}
Therefore, the area of triangle \\Delta DEF is \\tfrac{3}{2} square units.
Step 3: Relate the area of the triangle formed by midpoints to \\Delta ABC
For any triangle, a triangle formed by joining the midpoints of its sides (called the medial triangle) has an area exactly one-fourth of the original triangle.
Since \\Delta DEF is that medial triangle, we have:
\\Delta DEF = \\frac{1}{4} \\Delta ABC.
This implies
\\Delta ABC = 4 \\times \\Delta DEF.
Step 4: Determine the area of \\Delta ABC
We have:
\\Delta ABC = 4 \\times \\Delta DEF = 4 \\times \\frac{3}{2} = 6.
Hence, the area of the triangle \\Delta ABC is 6 square units.
Solution Diagram
